Controlled fermion mixing and FCNCs in a Δ(27) 3+1 Higgs Doublet Model
A. E. Cárcamo Hernández, Ivo de Medeiros Varzielas, M. L. López-Ibáñez, Aurora Melis
CControlled fermion mixing and FCNCs in a ∆(27)
A. E. C´arcamo Hern´andez a , ∗ Ivo de Medeiros Varzielas b , † M.L. L´opez-Ib´a˜nez c , ‡ and Aurora Melis d § a Universidad T´ecnica Federico Santa Mar´ıaand Centro Cient´ıfico-Tecnol´ogico de Valpara´ısoCasilla 110-V, Valpara´ıso, Chile, b CFTP, Departamento de F´ısica,Instituto Superior T´ecnico,Universidade de Lisboa,Avenida Rovisco Pais 1, 1049 Lisboa, Portugal c CAS Key Laboratory of Theoretical Physics,Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China. d Laboratory of High Energy and Computational Physics,National Institute of Chemical Physics and Biophysics,R¨avala pst. 10, 10143 Tallinn, Estonia (Dated: February 12, 2021)We propose a 3+1 Higgs Doublet Model based on the ∆(27) family symmetry supplementedby several auxiliary cyclic symmetries leading to viable Yukawa textures for the Standard Modelfermions, consistent with the observed pattern of fermion masses and mixings. The charged fermionmass hierarchy and the quark mixing pattern is generated by the spontaneous breaking of thediscrete symmetries due to flavons that act as Froggatt-Nielsen fields. The tiny neutrino massesarise from a radiative seesaw mechanism at one loop level, thanks to a preserved Z (1)2 discretesymmetry, which also leads to stable scalar and fermionic dark matter candidates. The leptonicsector features the predictive cobimaximal mixing pattern, consistent with the experimental datafrom neutrino oscillations. For the scenario of normal neutrino mass hierarchy, the model predicts aneffective Majorana neutrino mass parameter in the range 3 meV (cid:46) m ββ (cid:46)
18 meV, which is withinthe declared range of sensitivity of modern experiments. The model predicts Flavour ChangingNeutral Currents which constrain the model, for instance Kaon mixing and µ → e nuclear conversionprocesses, the latter which are found to be within the reach of the forthcoming experiments. I. INTRODUCTION
The Standard Model (SM) is unable to describe the observed pattern of SM fermion masses and mixings, which includesthe large hierarchy among its numerous Yukawa couplings. To address the flavour problem, a promising option is toadd family symmetries and obtain the Yukawa couplings from an underlying theory through the spontaneous breakingof the family symmetry.∆(27) as a family symmetry is greatly motivated by being one of the smallest discrete groups with a triplet andanti-triplet and the interesting interplay it has with CP symmetry. ∆(27) has been used in [1–34].We consider here a 3+1 Higgs Doublet Model (HDM) based on the ∆(27) family symmetry supplemented by severalcyclic symmetries, where three of the SU (2) doublets transform as an anti-triplet of ∆(27), H . The other doublet, h , ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] a r X i v : . [ h e p - ph ] F e b does not acquire a Vacuum Expectation Value (VEV) since it is charged under a preserved Z (1)2 and couples only tothe neutrino sector. Thus, the light active neutrino masses are generated from a radiative seesaw mechanism at oneloop level mediated by the neutral components of the inert scalar doublet h and the right handed Majorana neutrinos.Due to the preserved Z (1)2 symmetry, our model has stable scalar and fermionic dark matter (DM) candidates. Thescalar DM candidate is the lightest among the CP-even and CP-odd neutral components of the SU (2)-doublet scalar h . Furthemore, the fermionic DM candidate corresponds to the lightest among the right handed Majorana neutrinos.The DM constraints can be fulfilled in our model for an appropriate region of parameter space, along similar linesof Refs. [27, 35–51]. A detailed study of the implications of DM properties in our model goes beyond the scope ofthis paper and is therefore deferred for a future work. The masses and mixing of the charged fermions arise from H . Realistic masses and mixing require further sources of ∆(27) breaking [8, 12, 14] (this is not specific to ∆(27),see [52]). For this purpose, the model includes flavons (singlets under the SM) that are triplets of ∆(27) and acquireVEVs at a family symmetry breaking scale, assumed to be higher than the EW breaking scale, thus allowing themto decouple from the low-energy scalar potential.The Left-Handed (LH) leptons transform as anti-triplets of ∆(27), and the combination of charged lepton couplingsto H and neutrino couplings to h leads to a model with radiative seesaw and featuring the predictive and viablecobimaximal mixing pattern, which has attracted a lot of attention and interest by the model building communitydue to its predictive power to yield the observed pattern of leptonic mixing [25, 30, 32, 53–62].The quarks transform as singlets of ∆(27) but their masses still originate from Yukawa terms involving H and adominant flavon VEV. The symmetries allow also terms with subdominant flavon VEVs which do not contribute tothe masses but do produce the leading contribution to Yukawa couplings with the additional physical Higgs fields,and give rise to controlled Flavour Changing Neutral Currents (FCNCs).Distinguishing family symmetry models that have similar predictions for the Yukawa couplings is particularly relevant,and FCNCs are arguably the most reliable way to do so (see e.g. [29, 63–65] for some recent examples). In the presentmodel, we study the FCNCs mediated by the physical scalars in the leptonic and quark sectors in order to constrainthe parameter space, and find that in particular the muon conversion process and Kaon observables already constrainthis model.The layout of this paper is as follows. In Section II we describe the proposed model and we present its symmetry andfield content. Section III describes the low energy scalar potential and discusses the mass spectrum of the light scalarswhich play relevant roles in phenomenology. In Section IV we discuss the quark (IV A) and lepton (IV B) couplingsto the scalars, showing the respective Lagrangian terms, Yukawa matrices that arise after family symmetry breaking,and model’s fits to the observables. Section V analyses the constraints that arise from FCNCs in the context of thismodel. We conclude in Section VI. II. THE MODEL
We consider an extension of the SM with additional family symmetry, which is broken at a high scale. The fullsymmetry G of the model exhibits the following spontaneous symmetry breaking pattern: G = SU (3) C × SU (2) L × U (1) Y × ∆ (27) × Z (1)2 × Z (2)2 × Z (3)2 × Z ⇓ Λ SU (3) C × SU (2) L × U (1) Y × Z (1)2 ⇓ vSU (3) C × U (1) Q × Z (1)2 , (1) H h σ φ φ φ φ ∆ (27) , , Z (1)2 Z (2)2 Z (3)2 Z × Z (1)2 × Z (2)2 × Z (3)2 × Z where Λ is the scale of breaking of the ∆ (27) × Z (1)2 × Z (2)2 × Z (3)2 × Z discrete group, which we assume to be muchlarger than the electroweak symmetry breaking scale v = 246 GeV.The model includes four scalar SU (2) L doublets, three arranged as an anti-triplet of ∆(27), H , and h which is asinglet of ∆(27), does not acquire a VEV, and is charged under the unbroken Z (1)2 . The scalar sector is furtherextended, to include four flavons (SM singlets) ∆(27) triplets φ A and one ∆(27) trivial singlet σ which plays the roleof a Froggatt-Nielsen (FN) field. The FN field σ acquires a VEV at a very large energy scale, spontaneously breakingthe Z discrete group and then giving rise to the observed SM fermion mass and mixing hierarchy. Furthermore,the ∆(27) triplet φ is introduced to build the quark Yukawa terms invariant under the ∆(27) family symmetry.The remaining ∆(27) triplets φ , φ and φ are introduced in order to get a light active neutrino mass matrixfeaturing a cobimaximal mixing pattern, thus allowing to have a very predictive lepton sector consistent with thecurrent neutrino oscillation experimental data. The scalar assignments under the ∆ (27) × Z (1)2 × Z (2)2 × Z (3)2 × Z discrete group are shown in Table I. Here the dimensions of the ∆ (27) irreducible representations are specified by thenumbers in boldface and the different charges are written in additive notation. q L q L q L u R u R u R d R d R d R ∆ (27) , , , , , , , , , Z (1)2 Z (2)2 Z (3)2 Z -4 -2 0 4 2 0 4 3 3Table II: Quark assignments under the ∆ (27) × Z (1)2 × Z (2)2 × Z (3)2 × Z l L l R l R l R N R N R N R ∆ (27) ¯3 1 , , , , , , Z (1)2 Z (2)2 Z (3)2 Z × Z (1)2 × Z (2)2 × Z (3)2 × Z The role of the different cyclic groups is described as follows. The Z (3)2 symmetry is crucial for separating the ∆(27)scalar triplet φ participating in the quark Yukawa terms from the ones appearing in the neutrino Yukawa interactions.The Z (2)2 symmetry is necessary for shaping a cobimaximal texture of the light neutrino mass matrix, thus allowinga reduction of the lepton sector model parameters and at the same time allowing to successfully accommodate theneutrino oscillation experimental data. The preserved Z (1)2 symmetry allows the implementation of a radiative seesawmechanism at one loop level, providing a natural explanation for the tiny masses of the light active neutrinos andalso enabling stable DM candidates. Finally, the spontaneously broken Z symmetry shapes a hierarchical structureof the SM charged fermion mass matrices which is crucial for a natural explanation of the SM charged fermion massand quark mixing pattern.The fermion sector includes three SM singlets, Z (1)2 charged Right-Handed (RH) neutrinos N iR in addition to the SMfermions. All the fermions are arranged as trivial singlets of ∆(27) with the exception of the charged leptons fields,where the SU (2) L doublets l L transform as an anti-triplet and the l iR transform as specific non-trivial singlets. Thequark and lepton assignments under the ∆ (27) × Z (1)2 × Z (2)2 × Z (3)2 × Z discrete group are shown in Tables II andIII, respectively.We stress here that, thanks to the preserved Z (1)2 symmetry, the scalar and fermion sectors of our model containstable DM candidates. The scalar DM candidate is the lightest among the CP-even and CP-odd neutral componentsof the SU (2) scalar doublet h . The fermionic DM candidate corresponds to the lightest among the RH Majorananeutrinos. It is worth mentioning that in the scenario of a scalar DM candidate, it annihilates mainly into W W , ZZ , tt , bb and h SM h SM via a Higgs portal scalar interaction. These annihilation channels will contribute to the DM relicdensity, which can be accommodated for appropriate values of the scalar DM mass and of the coupling of the Higgsportal scalar interaction. Thus, for the DM direct detection prospects, the scalar DM candidate would scatter off anuclear target in a detector via Higgs boson exchange in the t -channel, giving rise to a constraint on the Higgs portalscalar interaction coupling. For the fermionic DM candidate, the lightest RH neutrino, the DM relic abundance canbe obtained through freeze-in, as shown in [27]. The DM constraints can therefore be fulfilled in our model for anappropriate region of parameter space, along similar lines of Refs. [27, 35–51, 66]. A detailed study of the implicationsof the DM candidates in our model is nevertheless beyond the scope of this work.With the particle content previously described, the scalar potential, as well as the Yukawa terms of up quarks, downquarks, charged leptons and the neutrino terms are constrained by the symmetries, which we consider in detail in thefollowing Sections. III. THE LOW ENERGY SCALAR POTENTIAL
The pattern of VEVs that we consider is (cid:104) H (cid:105) = v H (0 , , , (cid:104) φ (cid:105) = v (1 , , , (cid:104) φ (cid:105) = v (0 , , , (2) (cid:104) φ (cid:105) = v (cid:0) , ω, ω (cid:1) , (cid:104) φ (cid:105) = v (0 , , − , (cid:104) σ (cid:105) = v σ ∼ λ Λ , (3)with v H = v √ , being v = 246 GeV, and λ (cid:39) .
225 the Cabibbo angle. We do not consider here in detail the potentialterms that give rise to the flavon VEVs. The special ∆(27) VEV directions shown above and used in our model havebeen obtained in the literature in the framework of Supersymmetric models with ∆(27) family symmetry throughD-term alignment mechanism [2] or F-term alignment mechanism [19]. Such VEV patterns have also been derived innon-supersymmetric models and have shown to be consistent with the scalar potential minimization equations for alarge region of parameter space, as discussed in detail in [23, 25, 30] (see also [67, 68]).For the low energy scalar potential, we consider that the flavons have been integrated out, and write the scalarpotential in four parts V = V H + V h + V Hh + V breakingHh . (4)We write the ∆(27)-invariant potential for H in the notation of [67, 68] V ( H ) = − µ H (cid:88) i,α H iα H ∗ iα + s (cid:88) i,α,β ( H iα H ∗ iα )( H iβ H ∗ iβ )+ (cid:88) i,j,α,β (cid:2) r ( H iα H ∗ iα )( H jβ H ∗ jβ ) + r ( H iα H ∗ iβ )( H jβ H ∗ jα ) (cid:3) (5)+ (cid:88) α,β (cid:2) d (cid:0) H α H β H ∗ α H ∗ β + cycl. (cid:1) + h.c. (cid:3) , Figure 1: Loop corrected masses versus tree-level expressions and limits for the parameters of the scalar potential. Anuncertainty of 3 GeV is assumed in the numerical computation of the Higgs mass. The rectangular band corresponds to theallowed values of the 126 GeV SM like Higgs boson. where the Greek letters denote the SU (2) L indices. An equivalent way of writing V ( H ) where the ∆(27) invarianceis more transparent is shown in Appendix B.The potential for the unbroken Z (1)2 -odd field h is simply V h = µ h (cid:0) hh † (cid:1) + γ (cid:0) hh † (cid:1) , (6)whereas the terms mixing h and the ∆(27) triplet H are V Hh = α (cid:0) HH † (cid:1) , (cid:0) hh † (cid:1) + α (cid:0)(cid:0) Hh † (cid:1) (cid:0) H † h (cid:1)(cid:1) , , (7)and expand to V Hh = α (cid:16) H H † + H H † + H H † (cid:17) (cid:0) hh † (cid:1) + α (cid:104)(cid:0) H h † (cid:1) (cid:16) H † h (cid:17) + (cid:0) H h † (cid:1) (cid:16) H † h (cid:17) + (cid:0) H h † (cid:1) (cid:16) H † h (cid:17)(cid:105) . (8)We also consider higher order terms allowed by the symmetries, even though they are suppressed. In these, we findthe leading order contribution to the mass splitting between the CP-even and CP-odd neutral components of h , arisesfrom the terms: V breakingHh = κ (cid:2)(cid:0) H † h (cid:1) (cid:0) H † h (cid:1)(cid:3) S φ Λ + κ (cid:2)(cid:0) H † h (cid:1) (cid:0) H † h (cid:1)(cid:3) S φ Λ + h . c . . (9)We present these terms as the splitting of the masses is needed in order to obtain viable neutrino masses throughthe radiative seesaw mechanism (see Section IV B). Another invariant term arises by replacing φ by φ , but thatterm does not produce the effective mass term needed to yield the mass splitting between the CP-even and CP-oddneutral components of h . From the non-renormalizable scalar interactions given in Eq. (9), using the corresponding∆(27) breaking VEV, we obtain: V breakingHh = β (cid:104)(cid:16) H † h (cid:17) (cid:16) H † h (cid:17) + ω (cid:16) H † h (cid:17) (cid:16) H † h (cid:17) + ω (cid:16) H † h (cid:17) (cid:16) H † h (cid:17)(cid:105) + β (cid:104)(cid:16) H † h (cid:17) (cid:16) H † h (cid:17) + ω (cid:16) H † h (cid:17) (cid:16) H † h (cid:17) + ω (cid:16) H † h (cid:17) (cid:16) H † h (cid:17)(cid:105) + h . c ., (10)where β i ≡ κ i v / Λ.The electroweak symmetry is spontaneously broken by the non-zero VEV of the third component of the ∆(27) scalartriplet, H . After that, three electrically charged and seven neutral Higgs fields arise. The latter correspond to threeCP-even ( s , s , s ), two CP-odd ( p and p ) and two CP-mixed states ( h and h ). At tree-level, the light and heavyscalars and pseudoscalars, arising from the mixing of the neutral components of H and H , are degenerate in mass: m s , s = m p , p = v (cid:18) r + r − s ∓ d (cid:19) , (11)where the tadpole relation, dVdH = − µ H v + s v = 0 , (12)has been taken into account. As H gets the non-zero VEV, it dos not mix with the first and second componentsof H . As usual, its CP-odd and charged component are absorbed by the gauge bosons, which acquire masses, and aneutral massive scalar appears. We identify it with the SM-like Higgs boson of mass 125 GeV: m h SM ≡ m s = 2 s v . (13)The CP-mixed neutral states are related to the ∆(27) singlet, whose squared mass matrix is: M h = µ h + v ( α + α − β ) β v sin π β v sin π µ h + v ( α + α + β ) . (14)As follows from Eq. (14), the mixing between the scalar and pseudoscalar components of h is proportional to β and,therefore, negligible if µ h (cid:29) β v sin π/
3. At tree-level, the eigenmasses are m h ,h = µ h + v α + α ∓ β ) . (15)The phenomenology of the model is analysed by implementing it in SARAH 4.0.4 [69–74] and generating the cor-responding SPheno code [75, 76], through which the numerical simulation in Section V is performed. In particular,loop corrections are taken into account to compute the spectrum of the model. They are specially important forthe SM-like Higgs, s , whose mass is very sensitive to radiative corrections from other scalars. In Figure 1, theloop-corrected mass of this scalar is represented against some of the parameters of the scalar potential. The colouredregions correspond to the parameter space of our model. The green band reflects a theoretical uncertainty of 3 GeVthat we consider in the estimation of the mass. As it can be observed, the requirement of reproducing the 125 GeVmeasured value sets non-trivial limits on some of the masses and quartic couplings in Eqs.(5) and (6): µ h (cid:46) × GeV , s (cid:46) . , r (cid:46) . , r , d (cid:46) . . (16)The other parameters in Eqs.(7) and (9), which are not bounded by the mass of the SM-like Higgs, are varied in thegeneral range P i ∈ [0 . , . ] during the numerical scan. Within those intervals, the masses of the resulting spectrumare: m s , p ∼ <
275 GeV , m s , p ∼ <
350 GeV , m h , h ∼ < . (17) IV. FERMION MASSES AND MIXINGSA. Quark masses and mixings
In the quark sector, due to the fields transforming as ∆(27) trivial singlets, there are several terms as the nine possiblecombinations of ¯ q iL u jR and the nine of ¯ q iL d jR are allowed by the symmetries. The quarks must necessarily coupleto H because h is secluded to the neutrino sector through the unbroken Z (1)2 . We present first the quark terms thatinvolve H contracting with φ , which eventually lead to the quark mass terms when the scalars acquire the respectiveVEVs ( H acquiring a VEV in the third direction only): L ( Q ) Y = (cid:0) q L q L q L (cid:1) y ( U )11 σ Λ y ( U )12 σ Λ y ( U )13 σ Λ y ( U )21 σ Λ y ( U )22 σ Λ y ( U )23 σ Λ y ( U )31 σ Λ y ( U )23 σ Λ y ( U )33 (cid:16) φ ∗ (cid:101) H (cid:17) , u R u R u R (18)+ (cid:0) q L q L q L (cid:1) y ( D )11 σ Λ y ( D )12 σ Λ y ( D )13 σ Λ y ( D )21 σ Λ y ( D )22 σ Λ y ( D )23 σ Λ y ( D )31 σ Λ y ( D )23 σ Λ y ( D )33 σ Λ (cid:0) φ H (cid:1) , d R d R d R + h . c . . (19)The remaining quark terms have H coupling to φ or φ (instead of coupling to φ ): δ L ( Q ) Y = (cid:0) q L q L q L (cid:1) x ( U )11 σ Λ x ( U )12 σ Λ x ( U )13 σ Λ x ( U )21 σ Λ x ( U )22 σ Λ x ( U )23 σ Λ x ( U )31 σ Λ x ( U )23 σ Λ x ( U )33 (cid:88) r = S ,S ,A c ( U ) r Λ (cid:16) φ (cid:101) H (cid:17) r φ u R u R u R (20)+ (cid:0) q L q L q L (cid:1) x ( D )11 σ Λ x ( D )12 σ Λ x ( D )13 σ Λ x ( D )21 σ Λ x ( D )22 σ Λ x ( D )23 σ Λ x ( D )31 σ Λ x ( D )23 σ Λ x ( D )33 σ Λ (cid:88) r = S ,S ,A c ( D ) r Λ (cid:0) φ ∗ H (cid:1) r φ ∗ d R d R d R (21)+ ( φ → φ ) + h . c . , where the r subscript denotes the possible ∆(27) representation and (cid:16) φ (cid:101) H (cid:17) S ⊃ (cid:16) (cid:101) H , ω (cid:101) H , ω (cid:101) H (cid:17) v , (cid:16) φ (cid:101) H (cid:17) S ⊃ (cid:16) (cid:101) H , , (cid:17) v , (cid:16) φ (cid:101) H (cid:17) S ⊃ (cid:16) ω (cid:101) H + ω (cid:101) H , ω (cid:101) H + (cid:101) H , (cid:101) H + ω (cid:101) H (cid:17) v , (cid:16) φ (cid:101) H (cid:17) S ⊃ (cid:16) , (cid:101) H , (cid:101) H (cid:17) v , (22) (cid:16) φ (cid:101) H (cid:17) A ⊃ (cid:16) ω (cid:101) H − ω (cid:101) H , ω (cid:101) H − (cid:101) H , (cid:101) H − ω (cid:101) H (cid:17) v , (cid:16) φ (cid:101) H (cid:17) A ⊃ (cid:16) , (cid:101) H , (cid:101) H (cid:17) v . Similar products arise from ( φ ∗ H ) r with the conjugation ω ↔ ω . After symmetry breaking, these terms lead toanother contribution to the masses (which can be absorbed into the previous terms, as the structure is exactly thesame), but also to Yukawa couplings to the other components of H . In the absence of these terms, we would have inplace a Natural Flavour Conservation mechanism as only H couples to the quarks, and no FCNCs from the neutralscalars. But with these terms, we have Yukawa couplings to H and H . While they have the same overall texture asthe mass terms, they have different coefficients, and therefore are only approximately diagonalized when going to themass basis of the quarks. They are therefore a source of FCNCs which is controlled by the symmetries. Explicitly,the mass matrices and Yukawa couplings take the forms M U = v √ v Λ y ( U )11 λ y ( U )12 λ y ( U )13 λ y ( U )21 λ y ( U )22 λ y ( U )23 λ y ( U )31 λ y ( U )32 λ y ( U )33 , M D = v √ v Λ y ( D )11 λ y ( D )12 λ y ( D )13 λ y ( D )21 λ y ( D )22 λ y ( D )23 λ y ( D )31 λ y ( D )32 λ y ( D )33 λ , (23) Observable Model value Experimental value m u [MeV] 1 .
52 1 . ± . m c [GeV] 0 .
63 0 . ± . m t [GeV] 172 . . ± . m d [MeV] 2 .
88 2 . ± . m s [MeV] 55 . . ± . m b [GeV] 2 .
86 2 . ± . θ q . . ± . θ q . . +0 . − . sin θ q . . +0 . − . J q . × − (cid:0) . +0 . − . (cid:1) × − Table IV: Model and experimental values of the quark masses and CKM parameters. Y ( U ) H = ω λ ( U ) H x ( U )11 λ x ( U )12 λ x ( U )13 λ x ( U )21 λ x ( U )22 λ x ( U )23 λ x ( U )31 λ x ( U )32 λ x ( U )33 , Y ( D ) H = ω λ (D) H x ( D )11 λ x ( D )12 λ x ( D )13 λ x ( D )21 λ x ( D )22 λ x ( D )23 λ x ( D )31 λ x ( D )32 λ x ( D )33 λ , (24) Y ( U ) H = λ (U) H x ( U )11 λ x ( U )12 λ x ( U )13 λ x ( U )21 λ x ( U )22 λ x ( U )23 λ x ( U )31 λ x ( U )32 λ x ( U )33 , Y ( D ) H = λ (D) H x ( D )11 λ x ( D )12 λ x ( D )13 λ x ( D )21 λ x ( D )22 λ x ( D )23 λ x ( D )31 λ x ( D )32 λ x ( D )33 λ , (25) δY ( U ) H = ω λ (U) H x ( U )11 λ x ( U )12 λ x ( U )13 λ x ( U )21 λ x ( U )22 λ x ( U )23 λ x ( U )31 λ x ( U )32 λ x ( U )33 , δY ( D ) H = ω λ (D) H x ( D )11 λ x ( D )12 λ x ( D )13 λ x ( D )21 λ x ( D )22 λ x ( D )23 λ x ( D )31 λ x ( D )32 λ x ( D )33 λ , (26)where it is convenient to introduce the global effective couplings as λ (U , D) H = v v √ (cid:16) c ( U,D ) S − c ( U,D ) A (cid:17) , λ (U , D) H = v v √ (cid:20)(cid:16) c ( U,D ) S + c ( U,D ) A (cid:17) + v v (cid:21) , λ (U , D) H = v v √ c ( U,D ) S . (27)We note again that the textures are the same for the Yukawa couplings and the mass matrices, but with differentcoefficients, such that the Yukawa couplings to H and H are not diagonalized in the quark mass basis.The physical observables of the quark sector, i.e., the quark masses, CKM parameters and Jarskog invariant [77, 78]can be very well reproduced in terms of natural parameters of order one. This is shown in Table IV, which for eachobservable, compares the model value with the respective experimental value.The model values above are obtained from the following benchmark point: M U = − . . . . − . − . . . . GeV ,M D = . . − . − . i . . . − . . . GeV . (28) R e h , I m h R e h , I m h × ν i ν j N k N l × v ×× v φ × v φ v H × v H Figure 2: One-loop Feynman diagram contributing to the light active neutrino mass matrix. Here i, j, k, l = 1 , , v φ stands for either v , v or v . B. Lepton masses and mixings
In the lepton sector, the number of Yukawa terms is much smaller due to the assignments under ∆(27). The chargedlepton and neutrino Yukawa terms invariant under the symmetries of the model are given by: L ( l ) Y = y ( l )1 σ Λ (cid:0) l L H (cid:1) , l R + y ( l )2 σ Λ (cid:0) l L H (cid:1) , l R + y ( l )3 σ Λ (cid:0) l L H (cid:1) , l R + h . c . , (29) L ( ν ) Y = y ( ν )1 (cid:16) l L φ ∗ (cid:101) h (cid:17) , N R + y ( ν )2 (cid:16) l L φ ∗ (cid:101) h (cid:17) , N R (30)+ y ( ν )3 (cid:16) l L φ ∗ (cid:101) h (cid:17) , N R + y ( ν )4 (cid:16) l L φ ∗ (cid:101) h (cid:17) , N R + y ( ν )5 (cid:16) l L φ ∗ (cid:101) h (cid:17) , N R (31)+ m N N R N c R + m N N R N c R + m N N R N c R + m N (cid:0) N R N c R + N R N c R (cid:1) + h . c . , (32)where the dimensionless couplings in Eqs. (29)-(30) are O (1) parameters.From the charged lepton terms and the VEV pattern we consider (see Eq. (3)), we obtain a diagonal mass matrix: M l = m e m µ
00 0 m τ . (33)with the charged lepton masses given by: m e = y ( l )1 v v σ √ = y ( l )1 λ v √ , m µ = y ( l )2 v v σ √ = y ( l )2 λ v √ , m τ = y ( l )3 v v σ √ = y ( l )3 λ v √ , (34)where in a slight abuse of notation, we have absorbed the O (1) parameters of the VEVs into redefinitions of y ( l )1 , y ( l )2 and y ( l )3 in the expressions with λ . y ( l )1 , y ( l )2 and y ( l )3 are assumed to be real. As in the quark sector, the Lagrangian inEq.(29) gives rise to FCNCs through additional Yukawa couplings that arise with the other components of H , namely H and H . The entries are of the same size of those in Y ( l ) H but in different positions, thus Y ( l ) H = √ v m τ m e m µ , Y ( l ) H = √ v m µ
00 0 m τ m e . (35)In the neutrino sector, sorting out the products in Eq. (30-32), the Yukawa and Majorana mass matrices display the0following structures: Y ν = 1 √ y ( ν )3 v + y ( ν )1 v y ( ν )2 v + y ( ν )4 v ω y ( ν )3 v ω y ( ν )2 v y ( ν )5 v ω y ( ν )3 v ω y ( ν )2 v − y ( ν )5 v , M N = m N m N m N m N
00 0 m N . (36)After the spontaneous breaking of the discrete symmetries and of the electroweak symmetry, the following neutrinoYukawa interactions arise: L ( ν ) Y = z ( ν )1 ν L (cid:0) h R − i h I (cid:1) (cid:101) N R + z ( ν )2 (cid:0) ω ν L + ω ν L (cid:1) (cid:0) h R − i h I (cid:1) (cid:101) N R + z ( ν )3 ν L (cid:0) h R − i h I (cid:1) (cid:101) N R + z ( ν )4 (cid:0) ω ν L + ω ν L (cid:1) (cid:0) h R − i h I (cid:1) (cid:101) N R + z ( ν )5 ( ν L − ν L ) (cid:0) h R − i h I (cid:1) N R + m (cid:101) N (cid:101) N R (cid:101) N c R + m (cid:101) N (cid:101) N R (cid:101) N c R + m N N R N c R + h . c . , (37)with m h R = m Re[ h ] and m h I = m Im[ h ] , while (cid:101) N R , (cid:101) N R are the physical Majorana neutrino fields arising from thecombinations of N R and N R . They are given by: (cid:32) (cid:101) N R (cid:101) N R (cid:33) = (cid:32) cos β − sin β sin β cos β (cid:33) (cid:32) N R N R (cid:33) (38)where the mixing angle β takes the form tan 2 β = − m N / ( m N − m N ). The z ( ν ) i are the Yukawa parameters inthe basis of diagonal M N obtained by performing the rotation in Eq.(38). In that basis, the neutrino Yukawa matrix( Y ν → (cid:101) Y ν ) maintains the structure of Eq.(36) but with new entries determined by z ( ν ) i . The explicit expression for (cid:101) Y ν and the relation between the y ( ν ) i and z ( ν ) i parameters is given in Appendix C. Therefore, in the basis where the RHneutrinos are diagonal, the light active neutrino mass is obtained from the radiative seesaw mechanism as M ν ≡ π ) (cid:101) Y ν m (cid:101) N f m (cid:101) N f
00 0 m N f (cid:101) Y Tν , (39)with f k = f (cid:16) m h R , m h I , m (cid:101) N k (cid:17) , f = f (cid:16) m h R , m h I , m N (cid:17) , k = 1 , . (40)The loop function f takes the form: f (cid:16) m h R , m h I , m N R (cid:17) = m h R m h R − m N R ln (cid:32) m h R m N R (cid:33) − m h I m h I − m N R ln (cid:32) m h I m N R (cid:33) , (41)where m h R , m h I are given in terms of the parameters of the scalar potential in the entries (1,1) and (2,2) of Eq. (15).One can show that the resulting light active neutrino mass matrix in Eq. (39) can be parametrized as: M ν = a dω dωdω be iθ cdω c be − iθ , (42)where the exact relations between the effective parameters a, b, c, d, θ and the lagrangian parameters z ( ν ) i are given inAppendix C. Here, we stress that c can be expressed in terms of b and θ , and that all the effective parameters dependon the flavon VEVs.The physical observables of the neutrino sector, i.e., the three leptonic mixing angles, the CP phase and the neutrinomass squared splittings for the normal mass hierarchy (NH) can be very well reproduced, as shown in Table V, startingfrom the following benchmark point: a (cid:39) .
64 meV , b (cid:39) .
89 meV , c (cid:39) − .
79 meV , d (cid:39) (1 .
59 + i .
83) meV , θ (cid:39) . ◦ . (43)1 Figure 3: Correlation between neutrino observables around the benchmarkpoint. The star corresponds to the benchmark point considered in the text,whereas the dashed lines correspond to the experimental 1 σ ranges of [79].Figure 4: The effective Majorana neutrino mass parameter m ββ against m min = m ν . The green shadow is the allowed for NO scenarios. The dashedline corresponds to the future sensitivity expected from the nEXO experiment.The inner plot zooms in the correlation between m ββ and m min . Observable Modelvalue Neutrino oscillation global fit values (NH)Best fit ± σ [79] Best fit ± σ [80] 3 σ range [79] 3 σ range [80]∆ m [10 − eV ] 7 .
51 7 . +0 . − . . +0 . − . . − .
14 6 . − . m [10 − eV ] 2 .
56 2 . +0 . − . . +0 . − . . − .
65 2 . − . θ l [ ◦ ] 34 .
45 34 . ± . . +0 . − . . − . . − . θ l [ ◦ ] 8 .
59 8 . +0 . − . . ± .
12 8 . − .
94 8 . − . θ l [ ◦ ] 44 .
89 48 . +0 . − . . +0 . − . . − .
32 40 . − . δ CP [ ◦ ] 203 .
15 216 +41 − +27 − −
360 120 − Table V: Model and experimental values of the neutrino mass squared splittings, leptonic mixing angles, and CP-violatingphase. The experimental values are taken from Refs. [79, 80].
This shows that our predictive model successfully describes the current neutrino oscillation experimental data. As c depends on b and θ , we conclude that with only four effective parameters, i.e., a , b , d and θ , we can successfullyreproduce the experimental values of the six physical observables of the neutrino sector: the neutrino mass squareddifferences, the leptonic mixing angles and the leptonic CP phase. The correlations between neutrino observables aredepicted in Figure 3, while the value of θ is almost constant. To obtain this Figure, the lepton sector parameterswere randomly generated in a range of values where the neutrino mass squared splittings, leptonic mixing parametersand leptonic CP violating phase are inside the 3 σ experimentally allowed range. We note also that obtaining thecorrect scale for the light neutrino masses (and therefore, for the effective parameters) is implicitly setting a magnitudefor v , / Λ (cid:46) − .Another important lepton sector observable is the effective Majorana neutrino mass parameter of the neutrinolessdouble beta decay, which gives us information on the Majorana nature of neutrinos. The amplitude for this processis directly proportional to the effective Majorana mass parameter, which is defined as follows: m ββ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j U ek m ν k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (44)where U ej and m ν k are the the PMNS leptonic mixing matrix elements and the neutrino Majorana masses, respectively.Figure 3 displays m ββ as as function of the smallest of the light active neutrino masses m min , which for the normalmass hierarchy scenario corresponds to m min = m ν . The points displayed are all consistent with the experimentaldata with a χ < .
5. We find that our model predicts the effective Majorana neutrino mass parameter in the range m ββ (cid:46) (3 −
18) meV for the case of normal hierarchy. The new limit T νββ / ( Mo) ≥ . × yr on the half-life of0 νββ decay in Mo has been recently obtained [81]. This new limit translates into a corresponding upper bound on m ββ ≤ (300 − Xe 0 νββ half-life of T νββ / ( Xe) ≥ . × yr at 90% CL. This can be converted intoan exclusion limit on the effective Majorana neutrino mass between 5 . . V. CONSTRAINTS FROM FCNCS
Given the Yukawa couplings discussed in the previous Section for quarks and leptons, we have generically concludedthat FCNCs will be present in both sectors, mediated by physical neutral Higgs fields. In this Section, we discussin more detail how specific processes already act to constrain the parameter space of the model and highlight thenear-future experiments that will further act to probe the model. To this aim, we performed a numerical simulation3in SPheno considering the free input parameters of our model in the following intervalseffective parameters : v Λ ∈ [0 . , . , v i (cid:54) =3 Λ ∈ [10 − , . , (45)scalar potential : µ h ∈ [0 , ] GeV , r , , d, s , α , ∈ [0 . , , β , ∈ [0 . , v Λ , (46)quark sector : x ( U,D ) ij , c ( U,D ) S ,A ∈ ± [0 . , . , (47)neutrino sector : y ( ν )1 , , , , ∈ [0 . , . , m N , , , ∈ [10 − , ] GeV with m N < m N < m N < m N . (48)The most relevant Yukawa couplings are Y ( U,D ) H , , which we analyse through the effective coupling λ ( U,D ) H , of Eq. (27).Considering Eq. (27), upper bounds on this quantity reflect on the ratio of the flavon VEVs and the scale Λ. Dueto the top quark Yukawa coupling, coming from Eq. (23), we expect v / Λ ∈ [0 . , . λ ( U,D ) H , to imply a hierarchy between v (larger) and v , v (smaller). This hierarchy suppressesthe effective couplings of Eq. (27). Notice that the value of these couplings is not constrained by the model itself,although there is a dependence of v , / Λ on the scale of light neutrino masses which suggests λ ( U,D ) H , (cid:46) × − .In Figure 5 we show the dependence of the quark flavor violating observables b → s γ and ε K on λ ( U,D ) H , . The modelprediction for these observables is displayed through ratios to their respective SM values. The narrow horizontalband indicates the limit where the experimentally allowed SM-like values are safely recovered. Figure 5 shows that b → sγ would constrain the value of the couplings to be below λ ( U,D ) H , (cid:46) .
5. Very similar bounds, for simplicity notdisplayed in the figure, come from B d ( s ) → µ + µ − and B d → τ + τ − . Figure 5 also shows that a more constraininglimit comes from the CP violating observable ε K . We see that this observable would effectively restrict the couplingsto λ ( U,D ) H , (cid:46) .
1, although most of the points concentrate at λ ( U,D ) H , (cid:46) − . In these plots, the orange points areexcluded by this and other constraints, mostly the requirement to obtain the light neutrino masses.In the leptonic sector, among the Lepton Flavour Violating (LFV) processes, the muon to electron flavour violatingnuclear conversion µ − + N ( A, Z ) → e − + N ( A, Z ) (49)is known to provide a very sensitive probe of lepton flavour violation. Currently the best upper bound on the µ → e nuclear conversion comes from the SINDRUM II experiment [84] at PSI, using a Gold stopping target. This gives acurrent limit on the conversion rate of CR( µ − Au → e − Au) < × − .Searches for µ → e conversion at the Mu2e experiment [85] in FNAL and the proposed upgrade to COMET (Phase-II)experiment [86] in J-PARC would achieve a similar sensitivity and an upper limit of CR( µ − Al → e − Al) < × − ,that is four orders of magnitude below the present bound. In the long run, the PRISM/PRIME [87] is being designedto probe values of the µ → e conversion rate on Titanium, which is smaller by 2 orders of magnitude: CR( µ − Ti → e − Ti) < − .We focus here on the µ → e conversion because, contrary to the naive expectation of µ → e nuclear conversion beingproportional to µ → eγ , in our model we observe an interesting enhancement of the µ → e nuclear conversion detachedfrom other LFV processes like µ → e γ , τ → ( e, µ ) γ , µ → e and τ → e, µ ), which remain suppressed. In fact, fromthe couplings in Eqs. (29), µ → e can be generated already at tree-level through the exchange of a neutral scalar.Because of this, the impressive future sensitivity in this process will place significant constraints on the proposedmodel.As the process also involves quarks (inside the nuclei), we find it convenient to show the observable in terms of theeffective parameters λ ( U,D ) H , (already used in the previous Figures) in Figure 6. The orange points regions are alreadyexcluded by light neutrino masses, the observed value of b → s γ or ε K . The dashed horizontal lines show the futurelimits (as discussed above). We observe that most of the predicted points of the model reside in a window fullyaccessible to future experiments.4 Figure 5: Limits on the effective coupling λ ( U,D ) H , coming from theobservable b → sγ (top plot) and the CP observable ε K (bottom plot). Thehorizontal band indicates the allowed range. Points already excluded byquark observables are light yellow, only dark points are compatible withthe neutrino observables.Figure 6: The LFV observable µ → e nuclear conversion versus the effectivecoupling λ ( U,D ) H , . The colours correspond to those of Figure 5. Figure 7: The LFV observable BR( µ → eγ ) versus the RH-neutrino masses. Here we have taken into account thelower bound of [0 . −
1] GeV on the right handed Majorana neutrino masses arising from Big BangNucleosynthesis (BBN) [88]. The colours correspond to those of Figure 5.
Figure 7 on the other hand shows that, while it is in theory possible to constrain the values of the RH neutrino massesthrough µ → eγ such that it would eventually lead to lower bounds on M and M , in practice the values expectedin our model are too small to allow this process to effectively probe the parameter space. VI. CONCLUSIONS
In this work we presented a model based on the ∆(27) family symmetry, featuring a low energy scalar potential with3+1 SU (2) doublet scalars arranged as an anti-triplet ( H ) and trivial singlet ( h ) of the family symmetry. The latterdoes not acquire a Vacuum Expectation Value since it is charged under a preserved Z (1)2 symmetry, and is secluded inthe neutrino sector, where it leads to a radiative seesaw mechanism that produces the tiny masses of the light activeneutrinos.The quarks, being singlets of ∆(27), couple to H through ∆(27) invariant combinations of H and at least one of the∆(27) triplet flavons - one such combination giving rise to their masses and mixing through the third component H (identified with the Standard Model-like Higgs), and other combinations giving rise to Yukawa couplings to H , . Theextra physical scalars are mixtures of H , and have off-diagonal couplings to the quarks that are controlled by thesymmetries.The SU (2) doublet leptons are arranged like H as anti-triplets of ∆(27). The respective invariant combinations don’tinvolve the triplet flavons, and include couplings to H leading to the charged lepton masses and to H , yieldingFlavour Changing Neutral Currents, which are nevertheless controlled by the symmetries. The specific combination ofneutrino masses that originate from radiative seesaw and through invariants featuring ∆(27) triplet flavons producesthe cobimaximal mixing pattern.Our model successfully accommodates the experimental values of the quark and lepton (including neutrino) masses,mixing angles, and CP phases. Furthermore, the effective Majorana neutrino mass parameter is predicted to be in therange 3 meV (cid:46) m ββ (cid:46)
18 meV for the case of normal hierarchy. Most of the predicted range of values for the effectiveMajorana neutrino mass parameter is within the declared range 5 . − . µ → e nuclear conversion processes and Kaon mixing, which already restrict the modelparameter space, and that are generally predicted by the model to be in a range within the reach of future experiments.6 Acknowledgments
The authors thank Avelino Vicente for very useful discussions. A.E.C.H is supported by ANID-Chile FONDE-CYT 1170803. IdMV acknowledges funding from Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) through thecontract IF/00816/2015 and was supported in part by the National Science Center, Poland, through the HAR-MONIA project under contract UMO-2015/18/M/ST2/00518, and by FCT through projects CFTP-FCT Unit 777(UID/FIS/00777/2019), PTDC/FIS-PAR/29436/2017, CERN/FIS-PAR/0004/2019 and CERN/FIS-PAR/0008/2019which are partially funded through POCTI (FEDER), COMPETE, QREN and EU. MLLI acknowledges support fromthe China Postdoctoral Science Foundation No.2020M670475. AM acknowledges support by the Estonian ResearchCouncil grants PRG803 and MOBTT86, and by the EU through the European Regional Development Fund CoEprogram TK133 “The Dark Side of the Universe.”
Appendix A: The ∆(27) discrete group
The ∆(27) discrete group has the following 11 irreducible representations: one triplet , one anti-triplet and ninesinglets k,l ( k, l = 0 , , ), where k and l correspond to the charges of two Z and Z (cid:48) generators of this group,respectively [89]. The ∆(27) irreducible representations fulfill the following tensor product rules [89]: ⊗ = S ⊕ S ⊕ A ⊗ = S ⊕ S ⊕ A ⊗ = (cid:88) r =0 r, ⊕ (cid:88) r =0 r, ⊕ (cid:88) r =0 r, k,(cid:96) ⊗ k (cid:48) ,(cid:96) (cid:48) = k + k (cid:48) mod ,(cid:96) + (cid:96) (cid:48) mod (A1)Denoting ( x , y , z ) and ( x , y , z ) as the basis vectors for two ∆(27) triplets (or ), one finds:( ⊗ ) S = ( x y , x y , x y ) , ( ⊗ ) S = 12 ( x y + x y , x y + x y , x y + x y ) , ( ⊗ ) A = 12 ( x y − x y , x y − x y , x y − x y ) , (cid:0) ⊗ (cid:1) r, = x y + ω r x y + ω r x y , (cid:0) ⊗ (cid:1) r, = x y + ω r x y + ω r x y , (cid:0) ⊗ (cid:1) r, = x y + ω r x y + ω r x y , (A2)where r = 0 , , ω = e i π . Appendix B: Scalar potential
The scalar potential for the ∆(27) triplet H can be written in the form: V H = − µ H (cid:0) HH † (cid:1) , + ρ (cid:0) HH † (cid:1) , (cid:0) HH † (cid:1) , + ρ (cid:0) HH † (cid:1) , (cid:0) HH † (cid:1) , + ρ (cid:0) HH † (cid:1) , (cid:0) HH † (cid:1) , + ρ (cid:104)(cid:0) HH † (cid:1) , (cid:0) HH † (cid:1) , + h.c. (cid:105) . (B1)The following relations hold between the parameters in Eq. (5) and those in Eq. (B1): s ≡ ρ + ρ , r ≡ ρ − ρ , r ≡ ρ − ρ , d ≡ ρ − ω ρ . (B2)7 Appendix C: Neutrino mass parameters
The neutrino Yukawa matrix, in the basis of diagonal RH-neutrinos, is given by (cid:101) Y ν = Y ν R Tβ = 1 √ z ( ν )1 z ( ν )3 z ( ν )2 ω z ( ν )4 ω z ( ν )5 z ( ν )2 ω z ( ν )4 ω − z ( ν )5 , (C1)where R β refers to the rotation in Eq.(38) and the Yukawa parameters are: z ( ν )1 = 1 √ (cid:104)(cid:16) y ( ν )3 v Λ + y ( ν )1 v Λ (cid:17) cos β − (cid:16) y ( ν )2 v Λ + y ( ν )4 v Λ (cid:17) sin β (cid:105) ,z ( ν )2 = v √ (cid:16) y ( ν )3 cos β − y ( ν )2 sin β (cid:17) ,z ( ν )3 = 1 √ (cid:104)(cid:16) y ( ν )3 v Λ + y ( ν )1 v Λ (cid:17) sin β + (cid:16) y ( ν )2 v Λ + y ( ν )4 v Λ (cid:17) cos β (cid:105) , (C2) z ( ν )4 = v √ (cid:16) y ( ν )3 sin β + y ( ν )2 cos β (cid:17) ,z ( ν )5 = y ( ν )5 v √ . The relation between the light effective neutrino mass parameters in Eq. (42) and the lagrangian parameters z νi readsas a = (cid:16) z ( ν )1 (cid:17) m (cid:101) N π f + (cid:16) z ( ν )3 (cid:17) m (cid:101) N π f ,b = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω (cid:16) z ( ν )2 (cid:17) m (cid:101) N π f + ω (cid:16) z ( ν )4 (cid:17) m (cid:101) N π f + (cid:16) z ( ν )5 (cid:17) m N π f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,c = (cid:16) z ( ν )2 (cid:17) m (cid:101) N π f + (cid:16) z ( ν )4 (cid:17) m (cid:101) N π f − (cid:16) z ( ν )5 (cid:17) m N π f , (C3) d = z ( ν )1 z ( ν )2 m (cid:101) N π f + z ( ν )3 z ( ν )4 m (cid:101) N π f ,θ = arg ω (cid:16) z ( ν )2 (cid:17) m (cid:101) N π f + ω (cid:16) z ( ν )4 (cid:17) m (cid:101) N π f + (cid:16) z ( ν )5 (cid:17) m N π f , with f k as defined in Eq. (40). The system admits a solution as long as c = b (sin θ − √ θ ) / √ [1] G. C. Branco, J. M. Gerard, and W. Grimus, “GEOMETRICAL T VIOLATION,” Phys. Lett. (1984) 383–386.[2] I. de Medeiros Varzielas, S. F. King, and G. G. Ross, “Neutrino tri-bi-maximal mixing from a non-Abelian discretefamily symmetry,”
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