Controlled fermion mixing and FCNCs in a $\Delta(27)$ 3+1 Higgs Doublet Model

We propose a 3+1 Higgs Doublet Model based on the $\Delta(27)$ family symmetry supplemented by several auxiliary cyclic symmetries leading to viable Yukawa textures for the Standard Model fermions, consistent with the observed pattern of fermion masses and mixings. The charged fermion mass hierarchy and the quark mixing pattern is generated by the spontaneous breaking of the discrete symmetries due to flavons that act as Froggatt-Nielsen fields. The tiny neutrino masses arise from a radiative seesaw mechanism at one loop level, thanks to a preserved $Z_2^{\left( 1\right)}$ discrete symmetry, which also leads to stable scalar and fermionic dark matter candidates. The leptonic sector features the predictive cobimaximal mixing pattern, consistent with the experimental data from neutrino oscillations. For the scenario of normal neutrino mass hierarchy, the model predicts an effective Majorana neutrino mass parameter in the range $3$~meV$\lesssim m_{\beta\beta}\lesssim$ $18$ meV, which is within the declared range of sensitivity of modern experiments. The model predicts Flavour Changing Neutral Currents which constrain the model, for instance Kaon mixing and $\mu \to e$ nuclear conversion processes, 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 includes the large hierarchy among its numerous Yukawa couplings. To address the flavour problem, a promising option is to add family symmetries and obtain the Yukawa couplings from an underlying theory through the spontaneous breaking of the family symmetry. ∆ (27) as a family symmetry is greatly motivated by being one of the smallest discrete groups with a triplet and anti-triplet and the interesting interplay it has with CP symmetry. ∆ (27) has been used in .
We consider here a 3+1 Higgs Doublet Model (HDM) based on the ∆ (27) family symmetry supplemented by several cyclic symmetries, where three of the SU (2) doublets transform as an anti-triplet of ∆ (27), H. The other doublet, h, does not acquire a Vacuum Expectation Value (VEV) since it is charged under a preserved Z (1) 2 and couples only to the neutrino sector. Thus, the light active neutrino masses are generated from a radiative seesaw mechanism at one loop 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. The scalar 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 lines of Refs. [27,[35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. A detailed study of the implications of DM properties in our model goes beyond the scope of this 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 acquire VEVs at a family symmetry breaking scale, assumed to be higher than the EW breaking scale, thus allowing them to decouple from the low-energy scalar potential.
The Left-Handed (LH) leptons transform as anti-triplets of ∆ (27), and the combination of charged lepton couplings to H and neutrino couplings to h leads to a model with radiative seesaw and featuring the predictive and viable cobimaximal mixing pattern, which has attracted a lot of attention and interest by the model building community due to its predictive power to yield the observed pattern of leptonic mixing [25,30,32,[53][54][55][56][57][58][59][60][61][62].
The quarks transform as singlets of ∆ (27) but their masses still originate from Yukawa terms involving H and a dominant flavon VEV. The symmetries allow also terms with subdominant flavon VEVs which do not contribute to the 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][64][65] for some recent examples). In the present model, we study the FCNCs mediated by the physical scalars in the leptonic and quark sectors in order to constrain the parameter space, and find that in particular the muon conversion process and Kaon observables already constrain this model. The layout of this paper is as follows. In Section II we describe the proposed model and we present its symmetry and field content. Section III describes the low energy scalar potential and discusses the mass spectrum of the light scalars which play relevant roles in phenomenology. In Section IV we discuss the quark (IV A) and lepton (IV B) couplings to 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 this model. 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 full symmetry G of the model exhibits the following spontaneous symmetry breaking pattern:  where Λ is the scale of breaking of the ∆ (27) × Z 2 × Z 18 discrete group, which we assume to be much larger than the electroweak symmetry breaking scale v = 246 GeV. The Z 18 symmetry and the three additional Z 2 symmetries are distinguished by superscripts and commute with ∆ (27).
The model includes four scalar SU (2) L doublets, three arranged as an anti-triplet of ∆ (27), H, and h which is a singlet of ∆(27), does not acquire a VEV, and is charged under the unbroken Z 2 . The scalar sector is further extended, to include four flavons (SM singlets) ∆ (27) triplets φ A and one ∆(27) trivial singlet σ which plays the role of a Froggatt-Nielsen (FN) field. The FN field σ acquires a VEV at a very large energy scale, spontaneously breaking the Z 18 discrete group and then giving rise to the observed SM fermion mass and mixing hierarchy. Furthermore, the ∆(27) triplet φ 3 is introduced to build the quark Yukawa terms invariant under the ∆(27) family symmetry. The remaining ∆(27) triplets φ 123 , φ 23 and φ 1 are introduced in order to get a light active neutrino mass matrix featuring a cobimaximal mixing pattern, thus allowing to have a very predictive lepton sector consistent with the current neutrino oscillation experimental data. The scalar assignments under the ∆ (27) Table I. Here the dimensions of the ∆ (27) irreducible representations are specified by the numbers in boldface and the different charges are written in additive notation. q1L q2L q3L u1R u2R u3R d1R d2R d3R ∆ (27) 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 2 × Z18. Superscripts differentiate between the multiple Z2 symmetries.
The role of the different cyclic groups is described as follows. The Z symmetry is necessary for shaping a cobimaximal texture of the light neutrino mass matrix, thus allowing a reduction of the lepton sector model parameters and at the same time allowing to successfully accommodate the neutrino oscillation experimental data. The preserved Z (1) 2 symmetry allows the implementation of a radiative seesaw mechanism at one loop level, providing a natural explanation for the tiny masses of the light active neutrinos and also enabling stable DM candidates. Finally, the spontaneously broken Z 18 symmetry shapes a hierarchical structure of the SM charged fermion mass matrices which is crucial for a natural explanation of the SM charged fermion mass and quark mixing pattern.
The fermion sector includes three SM singlets, Z 2 charged Right-Handed (RH) neutrinos N iR in addition to the SM fermions. 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. The quark and lepton assignments under the ∆ (27) × Z 2 × Z 18 discrete group are shown in Tables II and III, respectively.
We stress here that, thanks to the preserved Z (1) 2 symmetry, the scalar and fermion sectors of our model contain stable DM candidates. The scalar DM candidate is the lightest among the CP-even and CP-odd neutral components of the SU (2) scalar doublet h. The fermionic DM candidate corresponds to the lightest among the RH Majorana neutrinos. 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 relic density, which can be accommodated for appropriate values of the scalar DM mass and of the coupling of the Higgs portal scalar interaction. Thus, for the DM direct detection prospects, the scalar DM candidate would scatter off a nuclear target in a detector via Higgs boson exchange in the t-channel, giving rise to a constraint on the Higgs portal scalar interaction coupling. For the fermionic DM candidate, the lightest RH neutrino, the DM relic abundance can be obtained through freeze-in, as shown in [27]. The DM constraints can therefore be fulfilled in our model for an appropriate region of parameter space, along similar lines of Refs. [27,[35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]66]. A detailed study of the implications of 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, down quarks, charged leptons and the neutrino terms are constrained by the symmetries, which we consider in detail in the following Sections.

III. THE LOW ENERGY SCALAR POTENTIAL
The pattern of VEVs that we consider is with v H = v √ 2 , being v = 246 GeV, and λ 0.225 the Cabibbo angle. We do not consider here in detail the potential terms that give rise to the flavon VEVs. The special ∆(27) VEV directions shown above and used in our model have been obtained in the literature in the framework of Supersymmetric models with ∆(27) family symmetry through D-term alignment mechanism [2] or F-term alignment mechanism [19]. Such VEV patterns have also been derived in non-supersymmetric models and have shown to be consistent with the scalar potential minimization equations for a large 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 scalar potential in four parts We write the ∆(27)-invariant potential for H in the notation of [67,68] where the Greek letters denote the SU (2) L indices. An equivalent way of writing V (H) where the ∆ (27) invariance is more transparent is shown in Appendix B.
The potential for the unbroken Z whereas the terms mixing h and the ∆(27) triplet H are and expand to We also consider higher order terms allowed by the symmetries, even though they are suppressed. In these, we find the leading order contribution to the mass splitting between the CP-even and CP-odd neutral components of h, arises from the terms: We present these terms as the splitting of the masses is needed in order to obtain viable neutrino masses through the radiative seesaw mechanism (see Section IV B). Another invariant term arises by replacing φ 123 by φ 1 , but that term does not produce the effective mass term needed to yield the mass splitting between the CP-even and CP-odd neutral components of h. From the non-renormalizable scalar interactions given in Eq. (9), using the corresponding ∆(27) breaking VEV, we obtain: where The electroweak symmetry is spontaneously broken by the non-zero VEV of the third component of the ∆(27) scalar triplet, H 3 . After that, three electrically charged and seven neutral Higgs fields arise. The latter correspond to three CP-even (s 0 1 , s 0 2 , s 0 3 ), two CP-odd (p 0 1 and p 0 2 ) and two CP-mixed states (h 0 1 and h 0 2 ). At tree-level, the light and heavy scalars and pseudoscalars, arising from the mixing of the neutral components of H 1 and H 2 , are degenerate in mass: where the tadpole relation, has been taken into account. As H 3 gets the non-zero VEV, it dos not mix with the first and second components of H. As usual, its CP-odd and charged component are absorbed by the gauge bosons, which acquire masses, and a neutral massive scalar appears. We identify it with the SM-like Higgs boson of mass 125 GeV: The CP-mixed neutral states are related to the ∆(27) singlet, whose squared mass matrix is: As follows from Eq. (14), the mixing between the scalar and pseudoscalar components of h is proportional to β 1 and, therefore, negligible if µ 2 h β 1 v 2 sin π/3. At tree-level, the eigenmasses are The phenomenology of the model is analysed by implementing it in SARAH 4.0.4 [69][70][71][72][73][74] and generating the corresponding 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 for the SM-like Higgs, s 0 3 , whose mass is very sensitive to radiative corrections from other scalars. In Figure 1, the loop-corrected mass of this scalar is represented against the parameters of the scalar potential in Eq (5): µ 2 h , s, r 1 , r 2 and d. The coloured regions correspond to the parameter space of our model. The green band reflects a theoretical uncertainty of 3 GeV that we consider in the estimation of the mass. As it can be observed, the requirement of reproducing the 125 GeV measured value sets non-trivial limits on some of the masses and quartic couplings in Eqs. (5) and (6): s 0.14, The other parameters in Eqs. (7) and (9), which are not bounded by the mass of the SM-like Higgs, are varied in the general range P i ∈ [0.05, 2.] during the numerical scan. Within those intervals, the masses of the resulting spectrum are:

A. Quark masses and mixings
In the quark sector, due to the fields transforming as ∆ (27) trivial singlets, there are several terms as the nine possible combinations ofq iL u jR and the nine ofq iL d jR are allowed by the symmetries. The quarks must necessarily couple to H because h is secluded to the neutrino sector through the unbroken Z 2 . We present first the quark terms that involve H contracting with φ 3 , which eventually lead to the quark mass terms when the scalars acquire the respective VEVs (H acquiring a VEV in the third direction only): The remaining quark terms have H coupling to φ 123 or φ 1 (instead of coupling to φ 3 ): where the r subscript denotes the possible ∆(27) representation and Similar products arise from (φ * 123 H) 3r with the conjugation ω ↔ ω 2 . After symmetry breaking, these terms lead to another contribution to the masses (which can be absorbed into the previous terms, as the structure is exactly the same), but also to Yukawa couplings to the other components of H. In the absence of these terms, we would have in place a Natural Flavour Conservation mechanism as only H 3 couples to the quarks, and no FCNCs from the neutral scalars. But with these terms, we have Yukawa couplings to H 1 and H 2 . While they have the same overall texture as the mass terms, they have different coefficients, and therefore are only approximately diagonalized when going to the mass 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 where it is convenient to introduce the global effective couplings as We note again that the textures are the same for the Yukawa couplings and the mass matrices, but with different coefficients, such that the Yukawa couplings to H 1 and H 2 are not diagonalized in the quark mass basis.
The quark masses and mixings are governed by the parameters y S2,A and they give subleading contributions to the SM quark mass matrices. Given the structure of the Yukawa couplings we do not consider our model to be predictive in the quark sector, beyond accounting for the hierarchies between the masses. 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 each observable, compares the model value with the respective experimental value.
The model values above are obtained from the following benchmark point:

B. Lepton masses and mixings
In the lepton sector, the number of Yukawa terms is much smaller due to the assignments under ∆ (27). The charged lepton and neutrino Yukawa terms invariant under the symmetries of the model are given by: 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: with the charged lepton masses given by: where in a slight abuse of notation, we have absorbed the O(1) parameters of the VEVs into redefinitions of y  In the neutrino sector, sorting out the products in Eq. (30)(31)(32), the Yukawa and Majorana mass matrices display the following structures: After the spontaneous breaking of the discrete symmetries and of the electroweak symmetry, the following neutrino Yukawa interactions arise: with are the physical Majorana neutrino fields arising from the combinations of N 1R and N 2R . They are given by: where the mixing angle β takes the form tan are the Yukawa parameters in the basis of diagonal M N obtained by performing the rotation in Eq. (38). In that basis, the neutrino Yukawa matrix (Y ν → Y ν ) maintains the structure of Eq.(36) but with new entries determined by z (ν) i . The explicit expression for 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 RH neutrinos are diagonal, the light active neutrino mass matrix is obtained from the radiative seesaw mechanism as shown in the Feynman diagram of Figure 2 and it is given by: with The loop function f takes the form: where the exact relations between the effective parameters a, b, c, d, θ and the lagrangian parameters z (ν) i are given in Appendix C. Here, we stress that c can be expressed in terms of b and θ, and that all the effective parameters depend on the flavon VEVs.
The masses of the charged leptons are set to the observed values, and the remaining parameters that govern the neutrino sector are y (ν) 1,2,3,4,5 , m N1,2,3,4 . The model is predictive as only the combinations a, b, c, d, θ of these parameters affect the physical observables of the neutrino sector, i.e., the three leptonic mixing angles, the CP phase and the neutrino mass squared splittings for the normal mass hierarchy (NH). These observables can be very well reproduced, as shown in Table V, starting from the following benchmark point: 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 successfully reproduce the experimental values of the six physical observables of the neutrino sector: the neutrino mass squared differences, the leptonic mixing angles and the leptonic CP phase. The correlations between neutrino observables are depicted in Figure 3, while the value of θ 23 is almost constant. To obtain this Figure, the lepton sector parameters were randomly generated in a range of values where the neutrino mass squared splittings, leptonic mixing parameters and leptonic CP violating phase are inside the 3σ experimentally allowed range. We note also that obtaining the correct scale for the light neutrino masses (and therefore, for the effective parameters) is implicitly setting a magnitude for v 123,1 /Λ 10 −2 .
Another important lepton sector observable is the effective Majorana neutrino mass parameter of the neutrinoless double beta decay, which gives us information on the Majorana nature of neutrinos. The amplitude for this process is directly proportional to the effective Majorana mass parameter, which is defined as follows: where U ej and m ν k are the PMNS leptonic mixing matrix elements and the neutrino Majorana masses, respectively.   data with a χ 2 < 1.5. We find that our model predicts the effective Majorana neutrino mass parameter in the range m ββ (3 − 18) meV for the case of normal hierarchy. The new limit T 0νββ 1/2 ( 100 Mo) ≥ 1.5 × 10 24 yr on the half-life of 0νββ decay in 100 Mo has been recently obtained [81]. This new limit translates into a corresponding upper bound on m ββ ≤ (300 − 500) meV at 90% CL. However, it is worth mentioning that the proposed nEXO experiment [82,83] will reach a sensitivity for the 136 Xe 0νββ half-life of T 0νββ 1/2 ( 136 Xe) ≥ 9.2 × 10 27 yr at 90% CL. This can be converted into an exclusion limit on the effective Majorana neutrino mass between 5.7 meV and 17.7 meV. In the most optimistic scenario this will exclude most of the predicted region of values of our model.

V. CONSTRAINTS FROM FCNCS
Given the Yukawa couplings discussed in the previous Section for quarks and leptons, we have generically concluded that FCNCs will be present in both sectors, mediated by physical neutral Higgs fields. In this Section, we discuss in more detail how specific processes already act to constrain the parameter space of the model and highlight the near-future experiments that will further act to probe the model. To this aim, we performed a numerical simulation in SPheno considering the free input parameters of our model in the following intervals scalar potential : quark sector : neutrino sector : y In Figure 5 we show the dependence of the quark flavor violating observables b → s γ and ε K on λ (U,D) H1,2 . The model prediction for these observables is displayed through ratios to their respective SM values. The narrow horizontal band 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  Figure 5 also shows that a more constraining limit comes from the CP violating observable ε K . We see that this observable would effectively restrict the couplings to λ 10 −2 (orange points). In these plots, the orange and yellow points are excluded by this and other constraints, mostly the requirement to obtain the light neutrino masses. In these plots, the requirement to obtain the light neutrino masses is very restrictive and only dark purple points satisfy all the constraints.
In the leptonic sector, among the Lepton Flavour Violating (LFV) processes, the muon to electron flavour violating nuclear conversion 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 a current limit on the conversion rate of CR(µ − Au → e − Au) < 7 × 10 −13 .
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) < 6 × 10 −17 , that is four orders of magnitude below the present bound. In the long run, the PRISM/PRIME [87] is being designed to probe values of the µ → e conversion rate on Titanium, which is smaller by 2 orders of magnitude: CR(µ − Ti → e − Ti) < 10 −18 .
We focus here on the µ → e conversion because, contrary to the naive expectation of µ → e nuclear conversion being proportional to µ → eγ, in our model we observe an interesting enhancement of the µ → e nuclear conversion detached from other LFV processes like µ → e γ, τ → (e, µ) γ, µ → 3 e and τ → 3 (e, µ), which remain suppressed. In fact, from the 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 proposed model.
As the process also involves quarks (inside the nuclei), we find it convenient to show the observable in terms of the effective parameters λ (already used in the previous Figures) in Figure 6. The orange (grey and lightest grey) points regions are already excluded by light neutrino masses, the observed value of b → s γ or ε K . The dashed horizontal lines show the future limits (as discussed above). Particularly from Figure 6, We observe that a large percentage of the predicted points of the model reside in a window accessible to future experiments. Figure 7 on the other hand shows that, while it is in theory possible to constrain the values of the RH neutrino masses through µ → eγ such that it would eventually lead to lower bounds on M 2 and M 3 , in practice the values expected in 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 with 3+1 SU (2) doublet scalars arranged as an anti-triplet (H) and trivial singlet (h) of the family symmetry. The latter does not acquire a Vacuum Expectation Value since it is charged under a preserved Z (1) 2 symmetry, and is secluded in the neutrino sector, where it leads to a radiative seesaw mechanism that produces the tiny masses of the light active neutrinos. The SU (2) doublet leptons are arranged like H as anti-triplets of ∆ (27). The respective invariant combinations don't involve the triplet flavons, and include couplings to H 3 leading to the charged lepton masses and to H 1,2 yielding Flavour Changing Neutral Currents, which are nevertheless controlled by the symmetries. The specific combination of neutrino masses that originate from radiative seesaw and through invariants featuring ∆(27) triplet flavons produces the 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 the range 3 meV m ββ 18 meV for the case of normal hierarchy. Most of the predicted range of values for the effective Majorana neutrino mass parameter is within the declared range 5.7 − 17.7 meV of sensitivity of modern experiments [83].
The detailed analysis of the Flavour Changing Neutral Currents lead to strong constraints on the model parameter   space. Of particular note are µ → e nuclear conversion processes and Kaon mixing, which already restrict the model parameter space, and that are generally predicted by the model to be in a range within the reach of future experiments.