Lepton mixing and the charged-lepton mass ratios

Journal of High Energy Physics, Mar 2018

Darius Jurčiukonis, Luís Lavoura

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Lepton mixing and the charged-lepton mass ratios

HJE mixing and the charged-lepton mass ratios Darius Jurˇciukonis 0 1 3 Lu´ıs Lavoura 0 1 2 0 1049-001 Lisboa , Portugal 1 Saule ̇tekio ave. 3, LT-10222 Vilnius , Lithuania 2 Instituto Superior T ́ecnico, CFTP, Universidade de Lisboa 3 Institute of Theoretical Physics and Astronomy, University of Vilnius We construct a class of renormalizable models for lepton mixing that generate predictions given in terms of the charged-lepton mass ratios. We show that one of those models leads, when one takes into account the known experimental values, to almost maximal CP -breaking phases and to almost maximal neutrinoless double-beta decay. We study in detail the scalar potential of the models, especially the bounds imposed by unitarity on the values of the quartic couplings. Phenomenological Models - Lepton 2 3 4 2.1 2.2 4.1 4.2 4.3 4.4 4.5 4.6 4.7 1 Introduction and notation Models A specific model The scalar potential Procedure for producing the scatter plots Scatter plots 5 Conclusions A Unitarity bounds for a 3HDM with Z2 × Z2 × Z2 symmetry A.1 General case A.2 Case with additional symmetry φ2 ↔ φ3 B Other stability points 1 Model (SM). The models have gauge group SU(2) × U(1) just as the SM. They feature an extended scalar sector, with three SU(2) doublets φk = φk , φ k + 0 T (k = 1, 2, 3) instead of one; we let φ˜k = φ0k∗, −φk − T denote the conjugate doublets. The leptonic sector is also extended, with the addition to the SM of three right-handed (i.e. SU(2)-singlet) neutrinos νRψ, 1 which enable a type-I seesaw mechanism [1–5] to suppress the standard-neutrino masses. Our models have family-lepton-number symmetries DLψ → eiξψ DLψ, ψR → eiξψ ψR, νRψ → eiξψ νRψ, (1.1) 1In this paper the Greek letters ψ, α, β, and γ in general run over the lepton flavours e, μ, and τ . Whenever we use (α, β, γ) we mean a permutation of (e, μ, τ ), i.e. α 6= β 6= γ 6= α. – 1 – where given by are the charged-lepton masses. The symmetries (1.1) leave the Yukawa couplings invariant but they are broken softly by the Majorana mass terms of the right-handed neutrinos, Mℓ = diag (ℓe, ℓµ , ℓτ ) , MD = diag (De, Dµ , Dτ ) , |ℓψ| = mψ LMaj = − 2 1 νeR, νµR , ντR  νeRT  MRC  νµR T  + H.c. ντRT  where the phases ξe, ξµ , and ξτ are arbitrary and uncorrelated. In transformation (1.1), ψR denotes the right-handed charged leptons and DLψ = (νLψ, ψL) doublets of left-handed neutrinos νLψ and charged leptons ψL. In our models both the charged-lepton mass matrix Mℓ and the neutrino Dirac mass matrix MD are diagonal, because they originate in Yukawa couplings that respect the family-lepton-number sym T denotes the SU(2) metries (1.1). Thus, In equation (1.4), C is the charge-conjugation matrix in Dirac space; the 3×3 flavour-space matrix MR is symmetric. The seesaw mechanism produces an effective light-neutrino mass matrix M = −MDMR−1MD, i.e. Mψψ′ = −DψDψ′ MR−1 ψψ′ , ∀ ψ, ψ′ ∈ {e, µ, τ } . Note that, since MD and Mℓ are diagonal, the matrix MR is the sole origin of lepton mixing in our models [6, 7].2 The symmetric matrix M is diagonalized as U T M U = diag (m1, m2, m3) , where the mk are the (non-negative real) light-neutrino masses and U is the lepton mixing matrix, for which we use the standard parameterization [9] (1.2) (1.3) (1.4) (1.5) (1.6) (1.7) HJEP03(218)5  c12c13 s12c13 U =  −s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ × diag 1, eiα21/2, eiα31/2 , where cij = cos θij and sij = sin θij for ij = 12, 23, 13. The phase δ is the Dirac phase; α21 and α31 are the Majorana phases. 2In the study of leptogenesis one uses a basis for the mass matrices where Mℓ and MR are diagonal but MD is not. In our models Mℓ and MD are diagonal but MR is not. If in our models MR is diagonalized as V T MRV = diag (M1, M2, M3), where V is a 3 × 3 unitary matrix and M1,2,3 are non-negative real, then M D′ = V T MD in the basis appropriate for the computation of leptogenesis. The Hermitian matrix relevant for leptogenesis is R ≡ M D′M D′† = V T MDM D†V ∗, which is non-diagonal. Thus, leptogenesis is in principle viable in our models. See ref. [8] for details. – 2 – The purpose of our models is to make predictions for the matrix U . There are in the literature many predictive models for U ;3 the models in this paper are original in that they are well-defined renormalizable models that produce predictions for the neutrino mass matrix M in terms of charged-lepton mass ratios. Since the mass ratios mµ /mτ , me/mµ , and me/mτ are very small, the predictions of our models are hardly distinguishable in practice from the cases with ‘texture zeroes’ in the neutrino mass matrix [ 13 ]. In section 2 we expound the construction of the models and classify the various models that our class of models encompasses. Section 3 focusses on a specific model with remarkable predictions: almost-maximal δ and almost-maximal neutrinoless double-beta decay. Section 4 discusses a scalar potential for our models and the way in which that potential is able to reproduce the Higgs particle discovered at the LHC. Section 5 contains our main conclusions. Appendix A deals on the derivation of the unitarity bounds on the coupling constants of the scalar potential. In appendix B we compute the expectation value of the scalar potential in the various stability points of that potential. 2 2.1 Models Construction of the models 1. Models with Yukawa Lagrangian Our class of models may be divided into four subclasses: where (α, β, γ) is a permutation of (e, µ, τ ) and y1,2,3,4 are Yukawa coupling constants, which are in general complex. 2. Models with Yukawa Lagrangian L1 = −y1DLανRαφ˜1 − y2DLααRφ1 − y3 DLβνRβφ˜2 + DLγ νRγ φ˜3 − y4 DLββRφ2 + DLγ γRφ3 + H.c., L2 = −y1DLανRαφ˜1 − y2DLααRφ1 − y3 DLβνRβφ˜2 + DLγ νRγ φ˜3 − y4 DLββRφ3 + DLγ γRφ2 + H.c. Note that L1 and L2 differ only in their last lines. 3. Models with Yukawa Lagrangian L3 = −y1DLανRαφ˜1 − y2DLααRφ1 − y3DLβνRβφ˜2 − y3∗DLγ νRγ φ˜3 − y4DLββRφ2 − y4∗DLγ γRφ3 + H.c., where y1 and y2 are real while y3 and y4 are in general complex. 3See the reviews [10–12]; the original papers are in the bibliographies of those reviews. – 3 – (2.1) (2.2) (2.3) 4. Models with Yukawa Lagrangian L4 = −y1DLανRαφ˜1 − y2DLααRφ1 − y3DLβνRβφ˜2 − y3∗DLγνRγφ˜3 − y4DLββRφ3 − y4∗DLγγRφ2 + H.c., where once again y1 and y2 are real. The Lagrangians (2.3) and (2.4) differ in their last lines. It is clear that L1,2,3,4 enjoy the family-lepton-number symmetries (1.1). The Lagrangians (2.1) and (2.2) further enjoy the interchange symmetry The Lagrangians (2.3) and (2.4) are invariant under the CP symmetry φ1 (x) → φ1∗ (x¯) , αR (x) → K αRT (x¯) , φ2 (x) → φ3∗ (x¯) , βR (x) → K γRT (x¯) , φ3 (x) → φ2∗ (x¯) , γR (x) → K βRT (x¯) , νRα (x) → K νRαT (x¯) , νRβ (x) → K νRγT (x¯) , νRγ (x) → K νRβT (x¯) , DLα (x) → KDLαT (x¯) , DLβ (x) → KDLγT (x¯) , DLγ (x) → KDLβT (x¯) , where x ≡ (t, ~r) and x¯ ≡ (t, −~r); K ≡ iγ0C is the CP -transformation matrix in Dirac space. Moreover, in the last line of transformation (2.6), T DLψ ≡ νLψT ! ψLT . The CP transformation (2.6) interchanges the lepton flavours β and γ. The Lagrangians (2.1)–(2.4) necessitate additional symmetries to guarantee that each scalar doublet only couples to the desired lepton flavour. There is a large arbitrariness in the choice of the additional symmetries. In this paper we choose them to be for all four Lagrangians (2.1)–(2.4); and either for Lagrangians (2.1) and (2.3), or else Z(22) : φ2 → −φ2, βR → −βR, νRβ → −νRβ, Z(23) : φ3 → −φ3, γR → −γR, νRγ → −νRγ, Z(24) : φ2 → −φ2, γR → −γR, νRβ → −νRβ, Z(25) : φ3 → −φ3, βR → −βR, νRγ → −νRγ, for Lagrangians (2.2) and (2.4). The transformations (2.8) and either (2.9) or (2.10) form a Z2 × Z2 × Z2 symmetry. – 4 – (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) Let vk denote the vacuum expectation value (VEV) of φ0k. Then, from L1, (Mℓ)αα ≡ ℓα = y2v1, (Mℓ)ββ ≡ ℓβ = y4v2, (Mℓ)γγ ≡ ℓγ = y4v3, (MD)αα ≡ Dα = y1∗v1, (MD)ββ ≡ Dβ = y3∗v2, (MD)γγ ≡ Dγ = y3∗v3 for model 1. From the Yukawa Lagrangian (2.2), for model 2. From L3, for model 3. From the Yukawa Lagrangian (2.4), assume that they do not break either the interchange symmetry (2.5) of models 1 and 2 or the CP symmetry (2.6) of models 3 and 4. This means that, in models 1 and 2, (MR)ββ = (MR)γγ , (MR)αβ = (MR)αγ . Clearly, the symmetry (2.15) for the matrix MR is also valid for the matrix MR−1. Therefore, from equation (1.5), for models 1 and 2. This means that the rephasing-invariant phase Mββ = Mγγ Dβ Dγ 2 , Mαβ = Mαγ Dβ Dγ arg hMγγ (Mαβ)2 Mβ∗β Mα∗γ 2i = 0 in models 1 and 2. Additionally, from equations (2.11) and (1.3), for model 1; while, from equations (2.12) and (1.3), for model 2. Mββ Mγγ Mββ Mγγ = = m2β m2γ m2γ m2 β , , Mαβ Mαγ Mαβ Mαγ = = mβ mγ mγ mβ – 5 – (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) HJEP03(218)5 We conclude that model 1 makes three predictions for the effective light-neutrino mass matrix M : equations (2.17) and (2.18). Model 2 also makes three predictions: equations (2.17) and (2.19). In models 3 and 4, we assume that the CP symmetry (2.6) is not broken by the Majorana mass terms of the νR. This means that (MR)ββ = (MR∗ )γγ , (MR)αβ = (MR∗ )αγ , (MR)αα = (MR∗ )αα , (MR)βγ = (MR∗ )βγ (2.20) (2.21) (2.22a) (2.22b) in those models. Equations (2.20) are valid for MR−1 as weall as for MR, hence , Mαβ = Mα∗γ DαDβ Dα∗Dγ∗ Mαα = Mα∗α DαDα Dα∗Dα∗ , Mβγ = Mβ∗γ DβDγ Dβ∗ Dγ∗ for models 3 and 4. Equations (2.21) imply the following rephasing-invariant conditions on the matrix M : arg hMβ∗βMγ∗γ (Mβγ )2i = 0, arg Mα∗αMβ∗γ MαβMαγ = 0. Moreover, from equations (1.3) and (2.13) one derives equation (2.18), which is thus also valid for model 3; from equations (1.3) and (2.14) one derives equation (2.19), which thus applies to model 4. We conclude that model 3 makes four predictions for M : equations (2.22) and (2.18). Model 4 also makes four predictions: equations (2.22) and (2.19). 2.2 Classification of the models Our class of models encompasses twelve models, depending on whether one uses model 1, 2, 3, or 4 and depending on whether the flavour α is taken to be e, µ , or τ . (The flavours β and γ are treated symetrically in the models.) There is a distinction between the models with interchange symmetry (2.5) and the models with CP symmetry (2.6): the former lead to only one constraint (2.17) on the phases of the matrix elements of M , while the latter lead to the two constraints (2.22). The CP symmetry (2.6) is more powerful than the interchange symmetry (2.5). However, in practice the distinction between equation (2.17) and equations (2.22) is not very significant, because the charged-lepton mass ratios are so small that they force some M -matrix elements to be very close to zero, hence their phases do not matter much. We see from equations (2.18) and (2.19) that our twelve models may be classified in six types: i. Models that predict Mee Mµ = me2 mµ2 , Meτ Mµτ = me mµ Since me ≪ mµ , in these models one is close to the situation Mee = Meτ = 0, which is case A2 of ref. [ 13 ]. – 6 – ii. Models that predict iii. Models that predict to case A1 of ref. [ 13 ]. iv. Models that predict v. Models that predict vi. Models that predict Mee Mµ Mee Mττ Mee Mττ Mµ Mττ Mµ Mττ = = = = = mµ2 me2 me2 mτ2 mτ2 me2 mµ2 mτ2 mτ2 mµ2 , , , , , Meτ Mµτ = mµ me . Meµ Mµτ = me mτ . Meµ Mµτ Meµ Meτ Meµ Meτ = = = mτ me mµ mτ mτ mµ . ≈ 0 and Meµ ≈ 0. They are therefore close to case B3 of ref. [ 13 ]. Since me ≪ mµ these models predict Mµ ≈ 0 and Mµτ ≈ 0. According to ref. [13], Since me ≪ mτ these models predict Mee ≈ 0 and Meµ ≈ 0. They are therefore close This leads to Mττ ≈ 0 and Meτ ≈ 0, corresponding to case B4 of ref. [ 13 ]. We thus find that, out of our twelve models, four should be phenomenologically excluded. The other eight are viable; two of them approximately coincide in their predictions with case A1 of ref. [ 13 ], two other with case A2, two more with case B3, and the last two with case B4. We have made numerical simulations of all our models and they very much vindicate the above conclusions. We do not feel it worth presenting those numerical simulations in detail here. In the next section we focus solely on one model that in our opinion yields particularly interesting results. 3 A specific model In this section we deal on one of our models, which predicts Mµ Mττ Meµ Meτ = = mµ2 mτ2 mµ mτ arg hMττ (Meµ )2 Mµ ∗ (Me∗τ )2i = 0. – 7 – (2.24) (2.25) (2.26) (2.27) (2.28) (3.1a) (3.1b) (3.1c) Equations (3.1) are three predictions. This is not much; for instance, each of the cases with two texture zeroes of ref. [ 13 ] has four predictions, and there are models with as many as six predictions for M . So, one might think that the predictions (3.1) are of little practical consequence. This is not so, however. We use M = U ∗ diag (m1, m2, m3) U † and the parameterization of U in equation (1.7). We also use the experimental 3σ bounds [ 14 ] and either for a normal ordering of the neutrino masses, or 95% confidence level [15].4 for the inverted ordering of the neutrino masses. The phases δ, α21, and α31 are unknown, just as the overall scale of the neutrino masses; we represent the latter through msum ≡ m1 + m2 + m3. Strong cosmological arguments suggest that msum ≤ 0.25 eV at A quantity of especial importance is mββ ≡ |Mee| = m1c122c123 + m2s122c123eiα21 + m3s123ei(α31−2δ) . This quantity is relevant for neutrinoless double-beta decay, which should proceed with a rate approximately proportional to m2ββ. It is clear that mββ becomes maximal when for whatever value of the phase δ. In figure 1 we have plotted mββ as a function of msum, both when only the inequalities (3.2) and either (3.3) or (3.4) hold, and when furthermore the predictions (3.1) are enforced. The information in that figure is clear: the predictions (3.1) lead to almost maximal mββ, irrespective of the neutrino mass ordering. This of course happens because equations (3.6) hold. In figure 2 one observes that this is indeed so and that, moreover, the predictions (3.1) lead to δ ≈ 3π/2. Thus, our model firmly predicts the three phases δ, α21, and α31; the phase δ is predicted to be very close to 1.5π, and this agrees nicely with its 1σ-preferred experimental value [ 14 ]. 4A recent paper [ 16 ] claims that msum = 0.11 ± 0.03 eV. – 8 – (3.2a) (3.2b) (3.2c) (3.3a) (3.3b) (3.4a) (3.4b) (3.5) (3.6) trino masses and figure 1b is for an inverted ordering. The red points take into account only the experimental bounds (3.2)–(3.4); the blue points arise from the constraints (3.1). is for a normal ordering of the neutrino masses and figure 2b is for an inverted ordering. One moreover observes in figure 1 that our model does not tolerate very low neutrino masses, but goes well with almost-degenerate neutrinos: msum & 0.15 eV for both the normal and inverted neutrino mass spectra. This specific model does not just predict the Dirac and Majorana phases; it moreover predicts the quadrant of the angle θ23 and a correlation between that angle and msum. That is observed in figure 3. One sees that θ23 lies in the first quadrant when the neutrino mass ordering is normal, in the second quadrant when it is inverted. One also sees that θ23 is correlated with msum, with θ23 becoming ever closer to π/4 when msum grows. Figures 1 and 3 are very similar to analogous figures displayed in ref. [17] for case B3 of ref. [ 13 ]. That case is defined by Mµ = Meµ = 0,5 which of course means four pre5The paper of ref. [ 13 ] contains various two-texture-zero cases, in particular case B3 defined as Mµ = Meµ = 0. The cases are of course not full models. However, it was demonstrated in ref. [ 18 ] that any texture-zero mass matrix may result from a renormalizable model. normal ordering of the neutrino masses and figure 3b is for an inverted ordering. dictions for M (because both the moduli and phases of Mµ and Meµ are relevant). Our predictions (3.1) mean that our model features both |Mµ | ≪ |Mττ | and |Meµ | ≪ |Meτ |, and this is an approximation to case B3. As a matter of fact, we have explicitly checked that the two conditions (3.1a) and (3.1b) by themselves alone lead to almost the same allowed domains as in figures 1–3, and as in case B3 of ref. [ 13 ]. The two conditions (3.1a) and (3.1b) are in practice just as predictive as that case with four predictions. 4 4.1 The scalar potential Assumptions In this section we investigate a way in which our class of models with three Higgs doublets and various symmetries may (i) be extended to the quark sector, and (ii) produce scalar particles with masses and couplings in agreement with the phenomenology. The aim of our investigation is to demonstrate that this can be done; we do not explore the full set of options. Thus, in this section we make additional assumptions. We stress that the validity of the models expounded in section 2 is in general independent of the specific additional assumptions that we shall utilize in this section. Our main assumption is that there are no scalars besides the three Higgs doublets that have Yukawa couplings to the leptons. Therefore, v ≡ q|v1|2 + |v2|2 + |v3|2 = √ 2mW g ≈ 174 GeV, (4.1) where mW = 80.1 GeV is the mass of the W ± bosons and g is the gauge-SU(2) coupling constant. Our models have either an interchange symmetry (2.5) or the CP symmetry (2.6). Those symmetries are unbroken by the Majorana mass terms of the right-handed neutrinos, which have mass dimension three. Still, those symmetries may be broken by the quadratic (i.e. mass dimension two) terms of the scalar potential. We shall assume that this does not happen, i.e. that either the interchange symmetry (2.5) or the CP symmetry (2.6) are conserved by the quadratic terms of the scalar potential. The potential is thus symmetric under either φ2 ↔ φ3 or φ2 ↔ φ3∗. In this paper we shall only consider the potential invariant under φ2 ↔ φ3;6 in ref. [ 19 ] the potential invariant under φ2 ↔ φ∗3 has been studied. Besides, the models have additional symmetries (2.8) together with either (2.9) or (2.10).7 The symmetry (2.8) does not involve the νRψ and is therefore unbroken by LMaj. We shall assume that it is also unbroken by the scalar potential; thus, the potential is invariant under Z(1) : φ1 → 2 −φ1.8 Z(2,4) : φ2 → −φ2 and Z(3,5) : φ3 → −φ3 in the scalar sector, are softly broken by LMaj, 2 2 which is of dimension three; therefore, they must also be broken in the quadratic part of The symmetries (2.9) or (2.10), which read the potential. The potential therefore is The parameters µ 3 and λ8 are real because of the symmetry under φ2 ↔ φ3. We use the freedom of rephasing φ1 to set λ7 real too. 4.2 The vacuum We assume as usual that the vacuum does not break electromagnetic invariance, i.e. that the upper components of φ1,2,3 have zero VEV. The potential (4.2) may, at least for some values of its parameters, produce stability points with non-trivial relative phases among the VEVs. Those stability points are, unfortunately, hard to manipulate analytically. We shall neglect them and assume that the 6Our potential is therefore invariant under a Z2 symmetry. When that Z2 symmetry is spontaneously broken, the vacuum is two-fold degenerate. There is a minimum-energy field configuration that interpolates between the two different vacua; this is called a domain wall. The non-observation of domain walls definitely is a problem for our potential. However, we recall the reader that our analysis only purports to display a particularly simple and illustrative case; we claim our potential neither to be realistic nor to be unique. The validity of the models expounded in section 2 is independent of the specific scalar potential that we analyze in this section. 7The additional symmetries are largely arbitrary — in the construction of the models we might have chosen different additional symmetries to the same practical effect, viz. preventing each scalar doublet from having Yukawa couplings to more than one lepton doublet. Each specific additional symmmetry alters the scalar potential in a different way. Thus, in a sense the specific additional symmetries (2.8)–(2.10) constitute an assumption of this section. 8In ref. [ 19 ] a potential with quadratic terms φ†1φ2, φ†1φ3, and their Hermitian conjugates has been analyzed. The fit in this section does not allow for those terms, which break the symmetry Z(21). See also footnote 6. three VEVs vk ≡ 0 φ0k 0 are (relatively) real. The VEV of the potential is then The equations for vacuum stability are We want a vacuum state with v1 6= 0, because in our models one of the charged-lepton masses is proportional to |v1|. We also want the vacuum to have |v2| 6= |v3|, because in our models r ≡ |v2/v3| is equal to a ratio of charged-lepton masses. Fortunately, equations (4.5) have a solution with v1 6= 0 and v2 6= ±v3: µ 1 = −2λ1v12 − 2l3 v22 + v32 , µ 2 = −2l3v1 − 2λ2 v22 + v32 , 2 µ 3 = 2 (λ2 − l4) v2v3. Plugging equations (4.6) into equation (4.3), we obtain V0 = , and we use r = mµ /mτ (the results for either r = me/mµ or r = me/mτ are not qualitatively different). The angle β will be taken to lie in the first quadrant. In this way v1 and v3 are positive, but this represents no lack of generality. Only the relative sign of v2 and v3 matters, and we have found out that the best results are obtained when v2v3 is negative. (4.3) (4.4a) (4.4b) (4.5a) (4.5b) (4.5c) (4.6a) (4.6b) (4.6c) (4.7) (4.8a) (4.8b) (4.8c) We expand the neutral components of the doublets as where the fields ρk and ηk are real. Subsuming the terms of the potential quadratic in the Mφ  φ2++  , φ 3 0  + (2λ2 − λ4 − λ6 + 2λ8)  0 −v32 v2v3  , (4.10a) we find, by using equations (4.6), that In general, the matrices Mη and Mφ must have an eigenvector (v1, v2, v3) with eigenvalue zero, corresponding to the Goldstone bosons, hence they must be of form Mη,φ = aη,φ  −v1v2 0  + bη,φ   + cη,φ  0  v 2 2 0 In our specific case, due to the φ2 ↔ φ3 symmetry of V , the coefficients aη,φ = bη,φ. This has the important consequence that both Mη and Mφ are diagonalized by the orthogonal matrix where where v23 ≡ pv22 + v32 = v cos β. We find that We diagonalize Mρ as  v1/v 0 −v23/v  Ov =  v2/v v3/v23 v1v2/ (vv23)  , where Oρ is a real, orthogonal matrix. We order its columns in such a way that m2H1 ≤ m2H2 ≤ m2H3. The fields Hk = P3k′=1 ρk′ (Oρ)k′k are physical scalars with mass mHk . 4.4 The oblique parameter T the oblique parameter T is [20] F (x, y) ≡  x + y   0 2 xy − x − y ln x y ⇐ x 6= y, ⇐ x = y, T = 1 3 16πs2wm2W 3 + X 3 X k=2 k′=1 + 3 X k=1 where mZ is the Z-boson mass, mW is the W -boson mass, mH is the reference mass of the Higgs boson (which is taken to be 125 GeV), and s2w = 1 − m2W /m2Z . According to ref. [9], −0.04 < T < 0.20. 4.5 There are many possible ways of extending our models to the quark sector. If one envisages a model with the CP symmetry (2.6), then that symmetry must be broken spontaneously through v2 6= v3∗ and that breaking must be felt in the quark sector, because we know that there is CP violation in that sector; this can be achieved only if both scalar doublets φ2 and φ3 have Yukawa couplings to the quarks. In a model with the interchange symmetry (2.5), on the other hand, CP violation may proceed through complex Yukawa couplings and it is not necessary for φ2 and φ3 to couple to the quarks. Things then become much simpler because at tree level there are no flavour-changing neutral currents mediated by the neutral scalars and therefore the neutral scalars do not need to be so heavy. Thus, we extend the symmetry Z(21) of equation (2.8) as where the QLk are the gauge-SU(2) doublets of left-handed quarks. With this extended Z(21), the quarks only couple to φ1. The Yukawa couplings of the quarks are then given by Lquark Yukawa = χ mχ √2v1 −ρ1 + iη1γ5 χ − X ζ=d,s,b ζ mζ ρ1 + iη1γ5 √2v1 X χ=u,c,t  ϕ+ 1 X X +  v1 χ=u,c,t ζ=d,s,b = − X k=1 3 Hk (Oρ)1k √2v sin β  + G0 − A3 cot β √2v +   G+ −ϕ3+ cot β v    X χ=u,c,t X χ=u,c,t ζ  Vχζ χ (mχPL − mζ PR) ζ + H.c. mχ χχ + mχ χiγ5χ − X ζ=d,s,b X ζ=d,s,b  mζ ζζ  mζ ζiγ5ζ  (4.20a) (4.20b) X X χ=u,c,t ζ=d,s,b Vχζ χ (mχPL −mζ PR) ζ + H.c., (4.20c) where PR,L are the projectors of chirality, G0 is the neutral Goldstone boson, G± are the charged Goldstone bosons, A3 is a physical pseudoscalar with mass mA3, and ϕ physical charged scalars with mass mϕ3. Notice in lines (4.20a) and (4.20b) the absence of 3± are the flavour-changing couplings of the neutral scalars. 4.6 Procedure for producing the scatter plots The input for our scatter plots is β and the eight λp (p = 1, . . . , 8). In inequalities (4.21), where Θ is the step (Heaviside) function. the λp, which are derived in appendix A: In order for the potential not to break unitarity we impose the following conditions on In order for the potential to be bounded from below we require that the λp satisfy [21] The angle β is an input of our scatter plots. The VEVs v1,2,3 are determined from equations (4.8), where v is given by equation (4.1) and r = mµ /mτ . Then, µ 1, µ 2, and µ 3 are computed by using equations (4.6). The value of V0 is given by equation (4.7). We require V0 < 0. We also enforce a number of conditions related to the alternative stability points in appendix B: require V0 < V0(1±), where the quantities V0(1±) are given in equation (B.2). • If the quantities in the right-hand sides of equations (B.1) are both positive, then we V (5±) are given in equation (B.11). V0 < V0(3), where V0(3) is given in equation (B.5). • If the quantity in the right-hand side of equation (B.4) is positive, then we require V0 < V0(4), where V0(4) is given in equation (B.9). • If the quantity in the right-hand side of equation (B.8a) is positive, then we require • If the quantity in the right-hand side of equation (B.10) is positive (with either the plus or the minus sign), then we require V0 < V (5±) (with the same sign), where 0 • If the quantity in the right-hand side of equation (B.12) is positive and the inequality (B.14) is satisfied, then we require V0 < V0(6), with V0(6) given in equation (B.13). We compute the squared masses in equations (4.15). We construct Mρ in equa tion (4.11c) and diagonalize it according to equation (4.16). We assume that the lightest physical scalar, viz. H1, corresponds to the scalar particle discovered at LHC ; we therefore fit its mass mH1 to be 125 GeV. This fit is very precise, hence mH1 never needs to appear in our scatter plots. We require that the masses of the six additional scalars, i.e. mϕ2,3 , mA2,3 , and mH2,3 , are all larger than 150 GeV. We also require the parameter T , computed through equation (4.18), to lie in between −0.04 and +0.20 [9]. The particle discovered at LHC, which we interpret as our H1, couples to gauge-boson pairs, to the heavy quarks, and to the τ lepton with strengths close to the predictions of the SM. We hence derive the following constraints: • The strength of the coupling of H1 to gauge-boson pairs, divided by the strength of the coupling of the SM Higgs boson to gauge-boson pairs, is [20] |gZZ |,9 where In our scatter plots we demand that 0.9 < gquarks < 1.1. 9The important quantity is |gZZ|, not gZZ itself, because the sign of the first column of the matrix Oρ is arbitrary and physically meaningless, hence the sign of gZZ is also arbitrary. Alternatively, we may reason that the physical cross sections depend on the squared amplitudes, hence on gZ2Z, not on the amplitudes themselves. gZZ ≡ v 1 3 k=1 X vk (Oρ)k1 . |gZZ | > 0.9. gquarks ≡ sin β |gZZ | (Oρ)11 gZZ . Note that −1 ≤ gZZ ≤ 1, because gZZ is the scalar product of two unit vectors. The limit |gZZ | = 1 corresponds to H1 coupling to pairs of gauge bosons with exactly the same strength as the SM Higgs boson does. In our scatter plots we require • We observe in equation (4.20a) that H1 couples to the quarks with strength (Oρ)11 sin β times the strength of the coupling to the quarks of the SM Higgs boson. Since the sign of (Oρ)11 is physically meaningless but is correlated with the sign of gZZ , we define (4.24) (4.25) (4.26) • We use |v2/v3| = mµ /mτ ; this means that we are assuming that, in our specific model, it is the scalar doublet φ3 that couples to DLτ τR. Thus, there is a Yukawa coupling The modulus of the Yukawa coupling constant Υ of course is mτ /v3. Since H1 couples to τLτR with strength (O√ρ)231v Υ.Therefore, for H1 to couple to τ leptons 2 . The modulus of the coupling of the SM Higgs boson to τLτR is mτ with the same strength as the SM Higgs boson, one needs to have (Oρ)31 v3 ≈ 1/v. Defining we demand that 0.9 < gτ < 1.1. gτ ≡ (Oρ)31 v gZZ , v3 |gZZ | Furthermore, we see in equation (4.20c) that the physical charged scalars ϕ3± interact with the quarks in the same way as the charged scalars of the type-I two-Higgs-doublet model. Therefore, in our scatter plots we have borrowed the bounds in the tan β–mϕ3 plane given in figure 18 of ref. [22]. 4.7 Scatter plots In figure 4 we plot the mass of the lightest new scalar, i.e. of H2, against β. One sees that β must always be close to 45◦ and that β becomes ever more restricted when the new-scalar masses get higher. Also notice that mH2 cannot be much higher than 300 GeV. In figure 5 we plot tan β against the mass of the physical charged scalars ϕ3± that interact with the quarks. Also marked in figure 5, through a solid line, is the phenomenological scalars ϕ lower bound on the mass of ϕ3±, which we have taken from figure 18 of ref. [22]. That bound incorporates the constraints from Z → b¯b, ǫK , and ΔmBs; it guarantees that the charged 3± do not mediate excessively strong |ΔS| = 2 transitions through box diagrams. In figure 6 we plot the quantities defined in equations (4.26) and (4.29) against each other. They seem to be anti-correlated; the anti-correlation becomes more well-defined when the masses of all the new scalar particles are higher. In figure 7 we plot the eight parameters λp of the scalar potential. One observes that |λp| is never larger than 2 for p ∈ {1, 2, 7, 8}; for 3 ≤ p ≤ 6 the λp may be somewhat larger. In figure 8 we have plotted the quartic Higgs coupling gH4 against the cubic Higgs coupling gH3. These are the coefficients of the terms (H1)4 and (H1)3, respectively, in the Lagrangian; in the case of gH3 we have multiplied the coefficient of (H1)3 by gZZ /|gZZ | in order to take into account the possibility that the field H1 has the wrong sign. One sees that the three-Higgs coupling may be almost twenty times larger than in the SM. Also, that coupling may be zero or even negative, i.e. it may have a sign opposite to the one in the (4.27) (4.28) (4.29) mH1 = 125 GeV, higher than 150 GeV; green points have all those masses higher than 200 GeV, and magenta points have all of them higher than 250 GeV. for the colours is the same as in figure 4. The solid line and the dashed line are phenomenological ± bounds extracted from figure 18 of ref. [22]. figure 4. as in figure 4. The black cross indicates the values of gH4 and gH3 in the SM. SM. The four-Higgs coupling is always larger than the corresponding SM coupling; it may at most be 60% larger than in the SM. We point out that, in a general two-Higgs-doubletmodel, the three-Higgs coupling has less freedom (it may at most be ten times larger than in the SM) than in this model, while the four-Higgs coupling has much more freedom than in this model — it may have values from zero until almost fifteen times larger than in the SM [23]. Therefore, a measurement of gH3 — of the cubic interaction of the 125 GeV scalar — may produce a large surprise and even distinguish this three-Higgs-doublet model from the most general two-Higgs-doublet one. In figure 9 we have plotted |gZZ | against the quartic Higgs coupling and against the cubic Higgs coupling. Notice that, although in our search we have restricted |gZZ | to have values in the range from 0.9 to 1, we have ended up obtaining only points with |gZZ | > 0.94. This is because we have restricted all the scalar masses (except the one of H1) to be larger than 150 GeV; larger scalar masses require a larger |gZZ | because the values of |gZZ | approach unity when the masses of all the new scalars are higher — this is the decoupling limit. 5 Conclusions In this paper we have constructed various extensions of the SM that yield predictions for the effective light-neutrino Majorana mass matrix M given in terms of the charged-lepton mass ratios. We have produced twelve models Mαpq, where α ∈ {e, µ, τ } and p, q ∈ {1, 2}. Models Mα1q predict Mββ Mγγ = m2β m2γ , Mαβ Mαγ = mβ mγ (5.1) the same as in figure 4. The dashed vertical lines indicate the values of gH3 and gH4 in the SM. where α 6= β 6= γ 6= α, whereas models Mα2q predict Furthermore, models Mαp1 predict whereas models Mαp2 predict Mββ Mγγ = m2γ m2 β , Mαβ Mαγ = mγ mβ . arg hMγγ (Mαβ)2 Mβ∗β Mα∗γ 2i = 0, arg hMβ∗βMγ∗γ (Mβγ )2i = 0, arg Mα∗αMβ∗γ MαβMαγ = 0. Mµ Mττ = mµ2 mτ2 , Meµ Meτ = mµ mτ , (5.2) (5.3) (5.4a) (5.4b) (5.5) In practice, the conditions (5.3) or (5.4) are not so important; this is because conditions (5.1) or (5.2) mean that two matrix elements of M are relatively small and lead to our models being approximations to two-texture-zero cases. Thus, eight of our twelve models are able to correctly fit the data: • Models Mµ 1q for q = 1, 2, which are approximations to case A1. • Models Mτ1q for q = 1, 2, which are approximations to case A2. • Models Me1q for q = 1, 2, which are approximations to case B3. • Models Me2q for q = 1, 2, which are approximations to case B4. The four models Mµ 2q and Mτ2q are not compatible with the phenomenological data and are therefore excluded. We have emphasized that our models Me1q lead, just from the two conditions to a vast predictive power, viz. δ ≈ 3π/2, α21 ≈ 0, α31 ≈ π, and almost maximal neutrinoless double-beta decay for either a normal or an inverted neutrino mass spectrum. Moreover, the quadrant of θ23 is correlated with the type of mass spectrum and θ23 approaches π/4 when the neutrino masses increase. We have carefully worked out a scalar potential appropriate to our models Me1q. (With slight modifications and no qualitatively different results, the potential is also appropriate to models Mµ 1q and Mτ1q.) Our assumptions were the following: • There are only three Higgs doublets φ1,2,3. • There is an interchange symmetry φ2 ↔ φ3 that is not softly broken in the quadratic part of the scalar potential. HJEP03(218)5 • The potential has an unbroken symmetry under φ1 → −φ1. • The vacuum expectation values are real. • The symmetry φ1 → −φ1 is extended to the quark sector in such a way that only φ1 has Yukawa couplings to the quarks; the physical neutral scalars therefore have no flavour-changing Yukawa couplings. CP violation is hard, i.e. it originates in complex Yukawa couplings. • The particle with mass 125 GeV discovered at LHC is the lightest physical scalar. Through a careful simulation we have found the appropriate ranges for the various parameters of the scalar potential. The physical-scalar masses cannot be much higher than a few hundred GeV. Acknowledgments L.L. thanks Pedro M. Ferreira, Joa˜o Paulo Silva, and Igor Ivanov for useful discussions. D.J. thanks the Lithuanian Academy of Sciences for support through the project DaFi2017. The work of L.L. is supported by the Portuguese Funda¸ca˜o para a Ciˆencia e a Tecnologia through the projects CERN/FIS-NUC/0010/2015 and UID/FIS/00777/2013, which are partially funded by POCTI (FEDER), COMPETE, QREN, and the European Union. A Unitarity bounds for a 3HDM with Z2 × Z2 × Z2 symmetry A.1 General case We consider the most general three-Higgs-doublet model with Z(1) 2 × 2 × Z(23) symmetry, Z(2) where Z(21) : φ1 → −φ1; Z(22) : φ2 → −φ2; Z(23) : φ3 → −φ3. (A.1) It is immaterial in this appendix whether any of the symmetries (A.1) is softly broken; here we just deal with the quartic part of the potential Vquartic = Λ1 φ1φ1 † 2 + Λ2 φ2φ2 + Λ3 φ3φ3 2 + Λ4 φ†1φ1 φ†2φ2 + Λ5 φ†1φ1 φ†3φ3 + Λ6 φ†2φ2 φ†3φ3 where the letters a, . . . , f denote creation/destruction operators as well as the corresponding particles. The (non-)existence of vacuum expectation values is immaterial for the unitarity bounds, therefore we neglect them in the notation (A.3). We denote the Hermitianconjugate operators through bars: a† → a¯, b† → ¯b, and so on. Then, a ! b 2 where Λ1,...,9 are real and Λ10,11,12 are in general complex. We follow ref. [24] to compute HJEP03(218)5 the unitarity bounds on the parameters of the potential (A.2). For notational simplicity, we write (A.2a) (A.2b) (A.2c) (A.2d) (A.3) (A.4a) (A.4b) (A.4c) (A.4d) (A.4e) (A.4f) (A.4g) (A.4h) (A.4i) (A.4j) (A.4k) (A.4l) (A.4m) (A.4n) (A.4o) (A.5) We must consider all the 2 → 2 scatterings that various pairs of particles may suffer among themselves. For instance, the three states aa, cc, and ee may, at tree-level, scatter through a matrix Vquartic = Λ1 a¯a¯aa + ¯b¯bbb + 2a¯¯bab + Λ2 c¯c¯cc + d¯d¯dd + 2c¯d¯cd + Λ3 e¯e¯ee + f¯f¯f f + 2e¯f¯ef + Λ4 a¯c¯ac + ¯bd¯bd + a¯d¯ad + ¯bc¯bc + Λ5 a¯e¯ae + ¯bf¯bf + a¯f¯af + ¯be¯be + Λ6 c¯e¯ce + d¯f¯df + c¯f¯cf + d¯e¯de + Λ7 a¯c¯ac + ¯bd¯bd + a¯d¯bc + ¯bc¯ad + Λ8 a¯e¯ae + ¯bf¯bf + a¯f¯be + ¯be¯af + Λ9 c¯e¯ce + d¯f¯df + c¯f¯de + d¯e¯cf + Λ10 a¯a¯cc + ¯b¯bdd + 2a¯¯bcd + Λ1∗0 c¯c¯aa + d¯d¯bb + 2c¯d¯ab + Λ11 a¯a¯ee + ¯b¯bf f + 2a¯¯bef + Λ1∗1 e¯e¯aa + f¯f¯bb + 2e¯f¯ab + Λ12 c¯c¯ee + d¯d¯f f + 2c¯d¯ef + Λ1∗2 e¯e¯cc + f¯f¯dd + 2e¯f¯cd .  2Λ1 2Λ10 2Λ11   2Λ∗10 2Λ2 2Λ12  .  2Λ∗11 2Λ∗12 2Λ3 The scattering matrices of the states (ad, bc), (af, be), and (bc, de) are Λ4 Λ7 ! Λ7 Λ4 , Λ5 Λ8 ! Λ8 Λ5 , Λ6 Λ9 ! Λ9 Λ6 , respectively. The scattering matrices of the states ad¯, ¯bc , af¯, ¯be , and f¯c, d¯e are (A.6) (A.8) (A.9) (A.10a) (A.10b) (A.10c) (A.11) respectively. The scattering matrix of the states a¯b, cd¯, ef¯ is The scattering matrix of the states a¯a, ¯bb, c¯c, d¯d, e¯e, f¯f is  2Λ1 Λ7 Λ8   Λ4 + Λ7  Λ5 + Λ8 2Λ1 4Λ1 Λ4 Λ5 Λ4 + Λ7 Λ5 + Λ8 Λ4 + Λ7 Λ4 4Λ2 2Λ2 Λ6 Λ6 + Λ9 Λ4 Λ4 + Λ7 2Λ2 4Λ2 Λ6 Λ6 + Λ9 Λ5 + Λ8 Λ6 + Λ9 Λ5 Λ6 4Λ3 2Λ3 Λ5 Λ6 2Λ3 4Λ3 Λ5 + Λ8  Λ6 + Λ9     .  In order to guarantee unitarity, we must enforce the condition that the moduli of all the eigenvalues of these matrices (and of a few more analogous matrices) are smaller than 4π. After some effort we find that those eigenvalues are Λ4 ± Λ7, Λ4 ± 2 |Λ10| , Λ5 ± Λ8, Λ5 ± 2 |Λ11| , Λ6 ± Λ9, Λ6 ± 2 |Λ12| , Λ4 + 2Λ7 ± 6 |Λ10| , Λ5 + 2Λ8 ± 6 |Λ11| , Λ6 + 2Λ9 ± 6 |Λ12| , and the eigenvalues of the matrices (A.5), (A.8), and Λ4 2Λ10 ! 2Λ∗10 Λ4 , Λ5 2Λ11 ! 2Λ∗11 Λ5 , Λ6 2Λ12 ! 2Λ∗12 Λ6 , (A.7)   6Λ1 2Λ4 + Λ7 2Λ5 + Λ8   2Λ4 + Λ7 6Λ2 2Λ6 + Λ9  . 2Λ5 + Λ8 2Λ6 + Λ9 6Λ3  In our case there is an additional symmetry φ2 ↔ φ3 in the potential, and that simplifies things much. Comparing equations (4.2) and (A.2), we see that (A.12a) (A.12b) (A.12c) (A.12d) (A.12e) (A.12f) (A.12g) (A.12h) (A.13a) (A.13b) (A.13c) (A.14) (B.1a) (B.1b) The quantities (A.10) then become λ3 ± λ5, λ3 ± 2λ7, and the matrices (A.5), (A.8), and (A.11) become   2λ1 2λ7 2λ7   2Λ7 2λ2 2λ8  ,  2λ7 2λ8 2λ2  2λ1 λ5 λ5   λ5 λ6 2λ2  λ5 2λ2 λ6  ,   6λ1 2λ3 + λ5 2λ3 + λ5  6λ2  The matrices (A.14) are 2–3 symmetric and therefore their eigenvalues are easy to find. One thus obtains the quantities in the left-hand sides of inequalities (4.23). B Other stability points Besides the vacuum state given by equations (4.6) and (4.7), there are several other stability points of the potential. The vacuum state must have a lower value of the potential than all other stability points. Therefore we must consider as many stability points as we can and, for each of them, compute the expectation value of the potential. That is what we do in the following. 1. Equations (4.5) have solutions with v1 6= 0 and v3 = ±v2. They are v12 = 2 (λ2 + l4) µ 1 − 4l3 (µ 2 ± µ 3) , v22 = −2l3µ 1 + 2λ1 (µ 2 ± µ 3) . 8l32 − 4λ1 (λ2 + l4) Plugging v3 = ±v2 together with equations (B.1) into equation (4.3), one obtains V0 = V0(1±) 2. The point v1 = v2 = v3 = 0 has 3. If v2 = v3 = 0 but v1 6= 0, there is a stability point with yielding to be positive and V0 = V0(2) ≡ 0. , V0 = V0(3) ≡ − 4λ1 4. If v1 = 0 but v2 6= 0 and v3 6= 0, we may analytically entertain the possibility that the VEVs of φ02 and φ03 have a relative phase ϑ. We take in this case both v2 and v3 V0 = µ 2 v22 + v32 + λ2 v24 + v34 + (λ4 + λ6) v22v32 + 2µ 3v2v3 cos ϑ + 2λ8v22v32 cos (2ϑ). The stationarity equations are 0 = µ 3 sin ϑ + 2λ8v2v3 sin (2ϑ), 0 = µ 2v2 + µ 3v3 cos ϑ + 2λ2v23 + (λ4 + λ6) v2v32 + 2λ8v2v32 cos (2ϑ), 0 = µ 2v3 + µ 3v2 cos ϑ + 2λ2v33 + (λ4 + λ6) v22v3 + 2λ8v2v3 cos (2ϑ). 2 This leads to the following possibilities: (a) cos ϑ = ±1 and v3 6= v2. 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Darius Jurčiukonis, Luís Lavoura. Lepton mixing and the charged-lepton mass ratios, Journal of High Energy Physics, 2018, 152, DOI: 10.1007/JHEP03(2018)152