Phenomenology of Dark Matter from Flavor Symmetry
Abstract
We investigate a model in which Dark Matter is stabilized by means of a parity that results from the same nonabelian discrete flavor symmetry which accounts for the observed pattern of neutrino mixing. In our example the standard model is extended by three extra Higgs doublets and the parity emerges as a remnant of the spontaneous breaking of after electroweak symmetry breaking. We perform an analysis of the parameter space of the model consistent with electroweak precision tests, collider searches and perturbativity. We determine the regions compatible with the observed relic dark matter density and we present prospects for detection in direct as well as indirect Dark Matter search experiments.
pacs:
95.35.+d 11.30.Hv 14.60.z 14.60.Pq 12.60.Fr 14.80.Cp 14.60.St 23.40.BwIFIC/1103
I Introduction
The existence of nonbaryonic Dark Matter (DM) is well established by cosmological and astrophysical probes. However, despite the great experimental effort over many years, its nature still remains elusive. Elucidating the longstanding puzzle of the nature of dark matter constitutes one of the most important challenges of modern cosmology and particle physics.
The various observations and experiments, however, constrain some of its properties Bertone:2004pz ; Taoso:2007qk . Among the most important requirements a DM candidate is required to satisfy are neutrality, stability over cosmological time scales, and agreement with the observed relic density. While the neutrality of the particle is usually easy to accomodate in models, its stability in general is assumed in an adhoc fashion. From a particle physics point of view, the stability suggests the existence of a symmetry that forbids the couplings that would otherwise induce the decay. Typically, the most common way to stabilize the DM particle is to invoke a parity, an example of which is R parity in supersymmetry.
It would certainly be more appealing to motivate such a symmetry from a topdown perspective. Different mechanisms have been suggested to achieve this Hambye:2010zb , for instance using gauge symmetries Frigerio:2009wf ; Kadastik:2009dj ; Batell:2010bp to get for example Rparity in the MSSM from a Martin:1992mq ) symmetery, global symmetries, accidental symmetries Cirelli:2005uq or custodial symmetry.
A new mechanism of stabilizing the DM has been recently proposed in Ref. Hirsch:2010ru in which DM stability originates from the flavor structure of the standard model. Indeed the same discrete flavor symmetry which explains the pattern of neutrino mixing Schwetz:2008er can also stabilize the dark matter ^{2}^{2}2Models based on nonAbelian discrete symmetries but with a decaying dark matter candidate can be found for example in Ref.Kajiyama:2010sb .. This opens an attractive link between neutrino physics and DM ^{3}^{3}3Other mechanisms of relating DM and neutrinos include the majoron DM Berezinsky:1993fm ; Lattanzi:2007ux ; Bazzocchi:2008fh .; two sectors that show a clear need for physics beyond the Standard Model.
The model proposed in Ref. Hirsch:2010ru is based on an symmetry extending the Higgs sector of the SM with three scalar doublets. After electroweak symmetry breaking two of the scalars of the model acquire vacuum expectation values (vev) which spontaneously break leaving a residual . The lightest neutral odd scalar is then automatically stable and will be our DM candidate.
On the other hand, the fermionic sector is extended by four right handed neutrinos which are singlets of . Light neutrino masses are generated via a type I seesaw mechanism Minkowski:1977sc ; gellmann:1980vs ; yanagida:1979 ; mohapatra:1980ia ; schechter:1980gr , obey an inverted hierarchy with and vanishing reactor neutrino angle ^{4}^{4}4For a similar realisation see Meloni:2010sk .. For pioneer studies on the use of for neutrino physics see Babu:2002dz .
We study the regions in parameter space of the model where the correct dark matter relic density is reproduced and the constraints from accelerators are fullfilled. We then consider the prospects for direct dark matter detection in underground experiments. We show that the model can potentially explain the DAMA annual modulation data Bernabei:2010mq ; Bernabei:2000qi as well as the excess recently found in the COGENT experiment Aalseth:2010vx . We show that present upper limits on the spin independent DM scattering cross section off nucleons can already severely constrain the parameter space of the model. Indirect dark matter searches through astrophysical observations are not currently probing the model apart from some small regions of the parameter space where the dark matter annihilation cross section is enhanced via a BreitWigner resonance.
The paper is organized as follows: in Sec.II we present the model, in Sec. III the constraints from collider data are reviewed and in Section IV we study the viable regions of the parameter space. In Sec. V and VI we sketch the prospects for direct and indirect dark matter detection. Finally, we summarize our conclusions in Sec. VII.
Ii The Model
We now provide a concrete realization of dark matter based upon the flavor symmetry adopting the simple typeI seesaw framewok Minkowski:1977sc ; gellmann:1980vs ; yanagida:1979 ; mohapatra:1980ia ; schechter:1980gr . In order to be phenomenologically viable, the model should account not only for the mixing angles describing the observed pattern of neutrino oscillations Schwetz:2008er but also for the two independent square mass splittings characterizing the “solar” and “atmospheric” sectors. These points were taken into account in the model proposed in Hirsch:2010ru which we now adopt, and consider its phenomenological features in more detail. The matter fields are assigned to irreducible representations of the group of even permutations of four objects Hirsch:2010ru , which is isomorphic to the symmetry group of the tetrahedron, for more details see Appendix A. The standard model Higgs doublet is assigned to a singlet representation, while the three additional Higgs doublets transform as an triplet, namely . The model has in total four Higgs doublets, implying the existence of four CP even neutral scalars, three physical pseudoscalars, and three physical charged scalar bosons.
In the fermion sector we have four righthanded neutrinos; three transforming as an triplet , and one singlet . Quarks are assigned trivially as singlets, as a result of which there are no predictions on their mixing matrix, as it is difficult to reconcile quarks with the neutrino sector ^{5}^{5}5The formulation of a full theory of flavour is beyond the scope of this work and will be left for the future..
The lepton and Higgs assignments are summarized in table 1.
2  2  2  1  1  1  1  1  2  2  
The resulting leptonic Yukawa Lagrangian is:
(1)  
This way the field is responsible for quark and charged lepton masses, the latter automatically diagonal. The scalar potential is:
(2)  
where is the product of two triplets contracted into one of the two triplet representations of , see eq. (14),and is the product of two triplets contracted into a singlet representation of . In what follows we assume, for simplicity, CP conservation, so that all the couplings in the potential are real. For convenience we also assume in order to have manifest conservation of CP in our chosen basis ^{6}^{6}6To see this, consider for instance the coupling arising from the terms proportional to and in eq. (2). Since this coupling is real if and only if ..
The minimization of the scalar potential results in :
(3) 
where all vevs are real. This vev alignment breaks the group to its subgroup responsible for the stability of the DM as well as for the neutrino phenomenology Hirsch:2010ru . In Appendix B we comment on a possible embedding of the model into the grand unified group .
The stability of the DM
Since there are no couplings with charged fermions nor quarks because of the symmetry, the only Yukawa interactions of the lightest neutral component of are with the heavy singlet righthanded neutrinos. This state is charged under the parity that survives after the spontaneous breaking of the flavor symmetry. One finds that, as a consequence of this symmetry, the mass matrix for the neutral scalars is blockdiagonal, see eq. (28), so that the lightest neutral component of is not mixed with the two Higgs scalars that take vev, and . Thus quartic couplings will not induce decays for this DM candidate which is therefore stable and constitutes our DM candidate.
We now show the origin of such a parity symmetry.
As explained in Appendix A, the group , has two generators: , and , that satisfy the relations . In the three dimensional basis is given by
(4) 
is the generator of the subgroup of . The alignment breaks spontaneously to since is manifestly invariant under the generator:
(5) 
For a generic triplet irreducible representation of , , we have:
(6) 
The residual symmetry is defined as
(7) 
and the rest of the matter fields are even, because the singlet representation transforms trivially under , see Appendix A. This is the residual symmetry which is responsible for the stability of our DM candidate.
Neutrino phenomenology
Here we summarize the main results obtained in Ref.Hirsch:2010ru concerning the neutrino phenomenology. The model contains four heavy righthanded neutrinos so it is a special case, called (3,4), of the general typeI seesaw mechanism schechter:1980gr . After electroweak symmetry breaking, it is characterized by Dirac and Majorana massmatrix:
(8) 
where and are respectively proportional to , , and of eq. (1) and are of the order of the electroweak scale, while are assumed close to the unification scale. Light neutrinos get Majorana masses by means of the typeI seesaw relation and the lightneutrinos mass matrix has the form:
(9) 
This texture of the light neutrino mass matrix has a null eigenvalue corresponding to the eigenvector ^{7}^{7}7 Note that if we were to stick to the minimal (3,3)typeI seesaw scheme, with just 3 SU(2) singlet states, one would find a projective nature of the effective treelevel light neutrino mass matrix with two zero eigenvalues, hence phenomenologically inconsistent. That is why we adopted the (3,4) scheme. implying a vanishing reactor mixing angle and inverse hierarchy. The atmospheric angle, the solar angle and the two square mass differences can be fitted. The model implies a neutrinoless double beta decay effective mass parameter in the range 0.03 to 0.05 eV at 3 , within reach of upcoming experiments.
Notation
After electroweak symmetry breaking and the minimization of the potential we can write:
(10) 
The structure of the neutral and charged scalar mass matrices follows from the exact symmetry, which forbids mixings between particles with different parities and from CP conservation. As a result the charged and neutral scalar mass matrices decompose into twobytwo mass matrices which, after diagonalization, give the scalar mass spectrum of the model. In what follows, unprimed particles denote mass eigenstates. We refer to Appendix C for details about the mass matrices and for a complete description of the mass spectrum. The odd sector contains two real CP even scalars, and , two real CP odd scalars, and and four charged scalars, and . The even scalars consist of two real CP even scalars and , that we generically call ’Higgses’, a pseudoscalar and two charged scalars The masses of the and gauge bosons impose the relation where stands for the standard model value of the vev. We call the ratio of the two vevs: .
The DM candidate of the model corresponds to the lightest odd neutral spin zero particle which, for the sake of definiteness, we take as the CPeven state . We remind that the parameters of the model relevant for the DM phenomenology are the 15 couplings of the scalar potential, and the ratio of the vacuum expectation values Indeed the minimization of the scalar potential and electroweak symmetry breaking allow to recast the mass parameters and in terms of the couplings and Note that the couplings and the Majorana mass parameters in Eq. Ref.1 determine the neutrino mass matrix, they are not relevant for the dark matter phenomenogy. Before moving to the calculation of its relic abundance in the next section we consider the phenomenological constraints on the parameter space of the model.
Iii Phenomenological constraints
In order to find the viable regions in parameter space where to perform our study of dark matter, we must impose the following constraints to the model :

Electroweak precision tests
It is wellknown that the oblique parameters provide stringent constraints on theories beyond the Standard Model Peskin:1991sw . Concerning the and parameters, these receive negligible contributions from the scalars of the model Grimus:2008nb ; Barbieri:2006dq , hence we focus on the parameter. We compute the effect on induced by the scalars following Grimus:2007if and we impose the bounds from electroweak measurements Nakamura:2010zzi :
While this bound favors a light Standard Model Higgs boson, large Higgs masses are possible in the presence of new physics, such as our multiHiggsdoublet model. Indeed, a negative deviation of the electroweak parameter induced by a heavy Higgs can be compensated by a positive produced by new scalar particles of the model, so one can raise the Higgs mass up to GeV or so (see below) ^{8}^{8}8A similar situation holds, for instance, in the Inert Doublet Model Barbieri:2006dq ..
We have explicitly verified that we can choose the mass spectrum of the model in such a way that this constraint is always respected. We refer to Appendix D for more detail.

Collider bounds
Searches for supersymmetric particles at LEP place a lower bound on the chargino mass of 100 GeV. To be conservative we apply this constraint also to the masses of the charged scalars of our model, even if slightly lower masses, GeV, might still be consistent with LEP dataPierce:2007ut .
The bounds imposed by LEP II on the masses of the neutral scalars in our model are similar to those constraining the Inert Doublet Model, given in Ref.Lundstrom:2008ai . This analysis applies directly to the odd scalar sector of our model of particular interest for DM phenomenology, as it constrains the mass difference between and . The excluded region that we adopt is taken from Fig.7 of Ref.Lundstrom:2008ai .
The masses of the even neutral scalars are also constrained by LEP searches. Although a precise bound requires a detailed analysis of the even sector, we just impose a lower limit on these masses of 114 GeV, which is approximatively the LEP limit on the SM higgs mass.
We remark that lepton flavor violating processes are suppressed by the large righthanded neutrino scale.

Perturbativity and vacuum stability
The requirement of perturbativity imposes the following bounds on the Yukawa couplings of the model and
this leads to an upper bound on the masses of the scalars at GeV ^{9}^{9}9Note that this bound would not apply to the Inert Higgs Doublet model, which would potentially allow heavier dark matter masses. The constraint on is necessary in order to preserve a perturbative topYukawa coupling.
Finally we must impose the stability of the vacuum. The conditions ensuring that the potential is bounded from below are:
where
Iv Relic Density
The thermal relic abundance of is controlled by its annihilation cross section into SM particles. In Fig.1 and Fig.2 we show the Feynman diagrams for the most relevant processes. In order to study the viable regions of the model we perform a random scan over the 16dimensional parameter space ( and ) and compute the dark matter relic abundance using the micrOMEGAs packageBelanger:2010gh ; Belanger:2008sj . Using the mass relations in equations 32 we can trade some of the couplings for the scalar masses. This system of 10 linear equations allows us to trade only 8 of the Indeed, the texture of the mass matrix of the odd sector imposes the following constraints on the squared mass differences: We linearly sample on the 8 independent masses in such a way to fulfill the collider constraints discussed in Sec.III. The remaining 7 free and are linearly sampled inside their allowed ranges, specified in Sec.III. Then, we select those choices which satisfy the perturbativity constraints for all the 15 the vacuum stability bounds and for which the electroweak precision tests are satisfied. Finally, we choose only those models which provide a relic adundance consistent with the WMAP measurements:
In Fig. 3 we show the regions with a correct relic abundance in the plane DM mass () and the lightest Higgs boson mass . For dark matter masses well below the threshold, dark matter annihilations into fermions are driven by the schannel exchange of the Higgs scalars of the model, as shown in Fig. 1.
For GeV the annihilation crosssections are large enough to obtain the correct relic density for all DM masses up to the the threshold. At larger Higgs boson masses, annihilations into light fermions are suppressed so that the relic abundance is typically too large unless efficient coannihilations with the pseudoscalar or with occur. These processes, shown in Fig.1 for , take place only for small mass splittings between and or . Note that the possibility to coannihilate with the charged scalar is ruled out, since LEP data requires GeV. There is, however, a narrow window still allowed by LEPII limits on the  planeLundstrom:2008ai where arbitrarily small mass splitting can exist between and for GeV. For lighter dark matter particles, strong coannihilations can not be reconciled with LEPII constraints as these require GeV Lundstrom:2008ai . We can therefore exclude by cosmological observations the parameter region correspondig to (simultaneously) GeV and GeV, because the DM would be overabundant. This is clearly seen in Fig. 3. In contrast, the absence of points on the strip corresponding to the line is associated to the presence of the resonance, which would enhance the DM annihilation cross section giving a too small Dark Matter abundance.
For dark matter masses larger than , unless the dark matter candidate is very heavy GeV, the annihilation cross section into gauge bosons is typically too large so that can only be a subdominant component of the dark matter budget of the Universe. However, for certain combinations of masses and parameters, the annihilations into gauge bosons may be suppressed by a cancellation between the Feynman diagrams (Fig.2), leading to an acceptable relic density. Indeed, this happens for some points in Fig. 3 ^{10}^{10}10For the Inert Doublet model, these cancellations have been noted in Lundstrom:2008ai and have been recentely studied in detail in LopezHonorez:2010tb .. Ref. LopezHonorez:2010tb shows that these processes allow for dark matter masses up to GeV, just below the threshold for annihilations into top pairs. In our scan, the viable region of the parameter space extends only up to GeV. Solutions at higher masses cannot be excluded, though their scrutiny is rather involved due to the large number of parameters of the model.
We now turn to the region GeV. As noted in Hirsch:2010ru , a scalar dark matter candidate annihilating into massive vector bosons inherits the correct relic abundance for this value of the mass if all other annihilation processes are absent. This scenario could be realised in our model by tuning the Yukawa couplings in order to suppress the dark matter annihilation into Higgs scalars and fermions. However, since the dark matter mass arises entirely from the electroweak symmetry breaking sector, high dark matter masses tend also to require large scalar couplings. This argument suggests that solutions with a heavy dark matter candidate and small couplings with the other scalars would be rather finetuned and we do not consider this possibility any further^{11}^{11}11 Note also that in the range GeV annihilations into three body final states may play a role, as shown for the Inert Doublet model in Honorez:2010re .. In the next two sections we study the prospects for direct and indirect dark matter detection.
V Direct detection
Dark Matter can be searched for in underground detectors looking for nuclear recoils induced by dark matter scattering against the target material. The scalar dark matter candidate we are considering couples to quarks via Higgs boson exchange, leading therefore to pure spin independent (SI) interactions with the nucleons. In the left panel of Fig.4 we show the SI scattering cross section off proton for the models with a correct dark matter abundance. We note that a large region of the parameter space is ruled out by the constrained imposed by current dark matter direct detection experiments.
A positive signal of dark matter detection has been claimed by the DAMA collaboration Bernabei:2000qi . DAMA has reported a high statistical evidence for annual modulation of the event rate over 13 year cycles Bernabei:2000qi ; Bernabei:2010mq . These results have prompted many attempts to interpret the data in terms of dark matter interactions with nuclei. Assuming an elastic WIMP interactions off nuclei and for ”standard” astrophysical assumption on the local DM density and velocity distribution, the DAMA signal is in conflict with the null results reported by other experimentsSavage:2010tg ; Kopp:2009qt ; Andreas:2010dz . However astrophysical inputs are subject to large uncertainties (see e.g.Belli:1999nz ; Green:2010gw ; Fox:2010bz ) and, moreover, the detector response is not completely known (for example the role of channelling is still under debate Bozorgnia:2010zc ). We also note that there exist alternative particle physics scenarios where a compability between the DAMA signal and the results of other experiments can be obtained, e.g.Barger:2010gv ; TuckerSmith:2001hy ; Chang:2009yt .
Recently, a hint of a possible signal from dark matter has been reported by the COGENT experiment Aalseth:2010vx . In Fig.4 we show the combination of SI scattering cross section and DM masses wich can fit the DAMA and COGENT results. An excess of events over the expected background has also been found by the CDMS experiment Ahmed:2009zw , although with a low statistical significance and by CRESST Cresst , even if these results are still preliminary. Interestingly, all these possible signals point to the same region of the parameter space, favoring low dark matter masses GeV Savage:2010tg ; Kopp:2009qt ; Schwetz:2010gv ; Fornengo:2010mk . As seen in Fig.4, these possible hints of dark matter detection can be accomodated within our model.
Current upper bounds severely challenge a possible interpretation of the aforementioned anomalies in terms of dark matter SI interactions. Indeed, as shown in Fig.4, the bulk of the COGENT and DAMA regions are excluded by the constraints inferred by XENON100 Aprile:2010um and CDMS Ahmed:2009zw ; Ahmed:2010wy . Still, experimental uncertainties may significantly affect the upper limits obtained by direct detection experiments, expecially for low mass WIMPs (see Savage:2010tg ; Schwetz:2010gv ; Collar:2010ht ). For this reason a possible dark matter spinindependent interpretation of the excess, even if disfavored, cannot be excluded at present.
Vi Indirect detection
Dark Matter indirect detection experiments look for signatures of DM annihilation into photons, neutrinos and (anti) matter fluxes. The expected DM signals depend on the astrophysical details related to the DM density distribution in the region of observation. Particle physics enters in the determination of the DM mass, annihilation cross section and the rates into various annihilation channels. In Fig. 4 we show at small velocity, relevant for DM annihilations inside our galaxy, as a function of . For GeV remains close to the thermal value at freezeout, as expected for the swave DM annihilation into light fermions. At larger DM masses, the presence of coannihilations allows for much smaller values of . The solutions at correspond to DM masses just below the Higgs resonance. In this case the annihilation cross section at small velocities is boosted with respects to its values at the DM freezeout. This behaviour of close to a narrow BreitWigner resonance has been recently widely exploited in order to boost the annihilation signal so to explain the cosmicrays anomalies reported by the PAMELA collaboration Ibe:2008ye ; Guo:2009aj ; Feldman:2008xs ; Ibe:2009dx .
In order to sketch the prospects for indirect DM detection we show in Fig. 4 the constraints on imposed by the FermiLAT observations of the Draco dwarf spheroidal galaxy Abdo:2010ex and the FermiLAT measurements of the isotropic diffuse gammaray emission Abdo:2010dk . We caution that these upper bounds have been computed assuming DM annihilations into , therefore they would directly apply only for parameter choices in our model where this annihilation channel dominantes. Still, this happens in large part of the parameter space, and in particular at low dark matter masses, where the FermiLAT constraints are close to the predictions of the model. For a comparison of these bounds with those obtained for different annihiliation channels we refer the reader to the original references. Further constraints for different targets of observations are obtained in Ref. Ackermann:2010rg ; Papucci:2009gd ; Cirelli:2009dv ; Zaharijas:2010ca ; Anderson:2010hh
One sees from Fig. 4 that current bounds are not yet able to significantly constrain the model. However, the FermiLAT sensitivity is expected to improve considerably with larger statistics and for different targets of observations, see e.g. Abazajian:2010zb ; Cuoco:2010jb ; Zaharijas:2010ca ; Anderson:2010hh ). For example, in Fig. 4 we show the forecasted 5 years FERMILAT sensitivities from the isotropic diffuse gammaray emission Abazajian:2010zb . FermiLAT measurements should be able to test the model for low dark matter masses.
Vii Conclusions and discussion
We have studied a model where the stability of the dark matter particle arises from a flavor symmetry. The nonabelian discrete group accounts both for the observed pattern of neutrino mixing as well as for DM stability. We have analysed the constraints that follow from electroweak precision tests, collider searches and perturbativity. Relic dark matter density constraints exclude the region of the parameter space where simultaneously GeV and GeV because of the resulting overabundance of dark matter. We have also analysed the prospects for direct and indirect dark matter detection and found that, although the former already excludes a large region in parameter space, we cannot constrain the mass of the DM candidate. In contrast, indirect DM detection is not yet sensitive enough to probe our predictions. However, forecasted sensitivities indicate that FermiLAT should start probing them in the near future.
All of the above relies mainly on the properties of the scalar sector responsible for the breaking of the gauge and flavour symmetry. A basic idea of our approach is to link the origin of dark matter to the origin of neutrino mass and the understanding of the pattern of neutrino mixing, two of the most oustanding challanges in particle physics today. At this level one may ask what are the possible tests of this idea in the neutrino sector. Within the simplest scheme described in Ref. Hirsch:2010ru one finds an inverted neutrino mass hierarchy, hence a neutrinoless double beta decay rate accessible to upcoming searches, while giving no CP violation in neutrino oscillations. Note however that the connection of dark matter to neutrino properties depends strongly on how the symmetry breaking sector couples to the leptons.
Viii Acknowledgments
This work was supported by the Spanish MICINN under grants FPA200800319/FPA and MULTIDARK Consolider CSD200900064, by Prometeo/2009/091, by the EU grant UNILHC PITNGA2009237920. S. M. is supported by a Juan de la Cierva contract. E. P. is supported by CONACyT (Mexico).
Appendix A The group
All finite groups are completly characterized by means of a set of elements called generators of the group and a set of relations, so that all the elements of the group are given as product of the generators. The group consists of the even permutations of four objects and then contains elements. The generators are and with the relations , then the elements are .
1  1  1  1  1 

1  1  
1  1  
3  3  0  0 
has four irreducible representations (see Table 2), three singlets and and one triplet. The onedimensional unitary representations are obtained by:
(11) 
where . The product rule for the singlets are:
(12) 
In the basis where is real diagonal,
(13) 
one has the following triplet multiplication rules,
(14) 
where and .
Appendix B embedding of the model
In Hirsch:2010ru the quark sector has not been studied and it is assumed that quarks are generically singlets of in order to not couple to the DM. It is possible to extend such a model to the quarks by embedding it into the grand unified group . It is beyond our scope to give a complete granunified model and we only sketch a simple way to embed the model presented in Hirsch:2010ru into a grand unified group. The matter assignment is the following
SU(5)  10  10  10  1  1  

1  1  3  1 
where we have assumed three copies of tenmultiplets and three of fivemultiplets of to decribe the three flavours. The scalar assignment is
SU(5)  

1  1 
then the Lagrangian is given by
(15)  
(16)  
(17) 
The charged lepton and down quark mass matrix are diagonal with eigenvalues
(18) 
and the three charged lepton masses as well as the three down quark masses can be easily reproduced. The up quark mass matrix is
(19) 
where
(20) 
Given the structure of the up and down quark mass matrices, the quark mixing matrix is diagonal. While this may be regarded as a good first approximation, since quark mixing angles are small, clearly another ingredient is needed such as, possibly, radiative corrections or extra vectorlike quark states. A full fit of the quark sector observables within a unified extension incorporating the flavour symmetry is beyond the scope of our paper.
Appendix C Mass spectrum
To simplify the notation we define the following combinations of couplings:
(21)  
(22)  
(23)  
(24)  
(25)  
(26) 
The neutral scalars massmatrix in the basis is block diagonal because of the symmetry and CPconservation. It is given by:
(27) 
and the charged scalars mass matrix in the basis is:
(28) 
The matrices and have a vanishing eigenvalue corresponding respectively to the neutral and charged Goldstone bosons eaten up by the and gauge bosons. The diagonalization of the mass matrices goes as follows:
(29) 
with and
(30) 
The mixing angle between and is . The ratio of the vevs was parametrized by the angle that controls the mixing within the even sector for the charged scalars and pseudoscalars:
(31) 
We do not report the lenghty expression for . The mass spectrum reads then as:
(32)  
(33)  
(34)  
(35)  
(36)  
(37)  
(38)  
(39)  
(40)  
(41) 
Appendix D Oblique parameters
Following the notation of Grimus:2007if , the oblique parameter for the standard model extended by higgs doublets with hypercharge is :
(46)  
where , denote the masses of the charged scalars and , are the masses of the neutral ones, is the finestructure constant and the function is defined as ():
(47) 
We evaluate the and matrices for our model as: