When $\tan \beta$ meets all the mixing angles

Models with two-Higgs-doublets and natural flavor conservation contain $\tan \beta = v_2 / v_1$ as a physical parameter. We offer here a generalization of a recently proposed idea where only the Cabibbo angle, $\theta_\text{c} \simeq 0.22$, was related to $\tan \beta$ by virtue of the $\mathbb{D}_{4}$ dihedral symmetry group. The original proposal consisted of a massless first generation of quarks and no mixing with the third generation. In our case, through the addition of a third Higgs doublet with a small vacuum-expectation-value but very large masses, thus later decoupling, all quarks become massive and quark mixing is fully reproduced. In fact, all quark mixing angles are expressed in terms of $\tan \beta$ and one recovers trivial mixing in the limit $\beta \rightarrow 0$. We also explore the consequences in lepton mixing by adopting a type I seesaw mechanism with three heavy right-handed neutrinos.


I. INTRODUCTION
Minimal scalar extensions of the standard model (SM) tackle the possibility of having more than one fundamental scalar in Nature.However, as their construction does not necessarily involve consideration of flavor symmetries, in general they have a large amount of arbitrariness.A part of this arbitrariness is represented by basis-dependent parameters which by definition are non-physical.Interestingly, when flavor symmetries are invoked, some of these parameters survive and become physical.Take for example the two-Higgs-doublet model (2HDM) [1,2] in its most general scenario.Then consider both doublets, Φ 1 and Φ 2 , to have the same quantum numbers, thus making them identical.By allowing the neutral components of both scalar doublets to acquire vacuum expectation values (VEVs), in general the latter must fulfill the condition v 2 1 + v 2 2 = v 2 = (174 GeV) 2 .Instead of {v 1 , v 2 } it is equivalent to employ {v, β} with tan β = v 2 /v 1 .It could seem that the angle is physical and provides a measure to distinguish between the two identical scalar doublets.However, this quantity is basis-dependent, as the kinetic terms in the scalar sector are left invariant under global 2 × 2 unitary transformations, and any such linear combination is an equally valid choice; that is, there is no preferred basis.
A preferred basis is only singled out by first imposing a certain symmetry (gauge, global, or discrete).The general scalar potential then reduces to a particular form.In particular, when using the reflection symmetry Z 2 , the non-physical parameter tan β can then be defined with respect to this basis and thereby promoted to a physical parameter.Additionally, if symmetry-breaking effects are allowed, the identification of this parameter as physical gets more subtle.For a thorough discussion on the physical meaning of tan β, see Ref. [3].
In 2HDMs with Z 2 , there is natural flavor conservation [4,5], that is, absence of flavor-changing-neutralcurrents (FCNCs) at tree-and loop-level.Interestingly, the Yukawa interactions get parametrised by the corresponding Yukawa couplings and tan β.One could thus naturally wonder if fermion mixing has anything to do with this parameter.This possibility was realized only recently [6].There it was found that by enlarging the discrete symmetry Z 2 to D 4 ≃ Z 4 ⋊ Z 2 and by a judicious assignment of the quarks and the two scalars to irreducible representations of D 4 [7], then the Cabibbo angle, θ c ≃ 0.22, can be directly related to β as: The proposal in [6] is a first attempt where the first generation of quarks remains massless and there is no allowed mixing with the third quark generation.It is our goal here to offer a complete framework where all quarks are massive and their mixings are consistent with the most up-to-date global fits, and similarly for the leptons.This letter is organized as follows.In Sec.II we describe the model.In Sec.III we discuss the scalar potential and show how to produce hierarchical VEVs.In Secs.IV and V we show how the mixing angles can be related to tan β.We discuss the main features and consequences of the model in Sec.VI.Finally, our conclusions are stated in Sec.VII.Appendix A provides a concise description of the D 4 tensorial products.In contrast to Ref. [6], we do the following representation assignments in the quark sector: whereas in the lepton sector we assign: Our choice allows a later reinterpretation of the model as having the appearance of a 2HDM with softly-broken natural flavor conservation.Notice that we are also considering D 4 -assignments of the leptonic fields, which are not included in the work of Ref. [6], as is only focused on the description of the quark sector.
On the other hand, the scalar sector, which is composed of three Higgs doublets, has the assignments or similarly, Φ S ∼ 1 −+ .However, the doublet scalar Φ S could not have been assigned to 1 −− as the quark mixing angles get the wrong hierarchy.Note that the scalar doublet Φ S is necessary as by virtue of it we introduce mixing with the third generation and a non-zero mass for the first fermion family.The Yukawa Lagrangian for the quark sector is whereas for the lepton sector is

III. THE SCALAR POTENTIAL
The most general, renormalizable, and D 4 -symmetric scalar potential is given by Because of Hermiticity, all parameters except for λ 7 are real.However, a phase redefinition of the fields can absorb the phase of λ 7 as well.Therefore, the potential is also CP -symmetric.We assume µ 2 D < 0 while µ 2 S > 0, such that As long as the flavor symmetry is not broken, the minimum conditions: enforce v 1 = v 2 (the symmetric limit).However, if we wish to explore the case where one of the two VEVs in the flavor doublet is rather small, i.e. v 2 ∼ O(10 −2 −10 −1 ) v 1 , we need to softly-break the symmetry by introducing where we assume µ 2 1 , µ 2 2 real, with µ 2 1 < 0 and µ 2 2 > 0.Moreover, as we are also interested in inducing an even smaller VEV than v 2 in the flavor singlet Higgs isodoublet, Φ S , we need to add an extra soft-breaking term; the complete expression therefore reads This choice allows us to write the following relations If we consider λ k ∼ O(1), and µ D ∼ 50 GeV, µ 1,2 ∼ 100 GeV, µ S2 ∼ 100 GeV, and µ S ∼ 0.5 TeV then we expect v 1 ∼ 100 GeV, v 2 ∼ 10 GeV, v s ∼ 1 GeV.In other words, v 2 = ǫv 1 and v s = ǫ 2 v 1 with ǫ ∼ 10 −1 .Now, note that this hierarchy in the VEVs allows us to say that to a very good degree of approximation and we are still able to write With this definition we can reexpress the small VEVs in the following way where we have already considered v 1 ≃ v and To illustrate the order of the scalar masses we consider a simple benchmark scenario where λ 2,3,6,7,8 = 0, λ 4 = λ 5 , and λ 1,4,5 ∼ O(1).The scalar mass matrices then read The CP -odd and charged scalar mass matrices are consistent with only one massless state.It is then straightforward to obtain that the lightest scalar state is of the order of √ 2µ 1 ∼ 100 GeV while all the heavy states (neutral and charged) of the order of µ 2 > 125 GeV and µ S > 0.5 TeV, in agreement with all the current experimental constraints.

IV. QUARK MASSES AND MIXINGS
The mass matrices in the quark sector are We calculate the unitary transformations diagonalizing the mass matrices from the hermitian mass matrix product, where the choice of sign ± corresponds to the up and down quark matrix, respectively.It is clear from here that if the VEVs satisfy the hierarchy one gets approximately where the hierarchy in the quark mixing angles is fully reproduced and the mass spectrum generically reads Notice that when v s → 0, the up-and down-type quark mass matrices shown in Eq. ( 22) are diagonalised by the orthogonal transformation Hence, the leading contribution to quark mixing is obtained from from which one easily sees that Eq. ( 1) is satisfied up to a sign which has no physical relevance.

V. LEPTON MASSES AND MIXINGS
The Dirac mass matrices in the lepton sector are while the corresponding one for the right-handed neutrinos is Note that the charged lepton masses get similar expressions as those in Eq. ( 25).
Assuming that the right handed Majorana neutrinos have masses much larger than the electroweak symmetry breaking scale, one has that the light active neutrino masses are generated from a type I seesaw mechanism that gives rise to an effective Majorana mass matrix.Unfortunately, it is straightforward to show that no matter what assigments one makes between the D 4 -irreps and the lepton fields, the model is incapable of fully reproducing the large leptonic mixing angles.However, one possibility to get out of this situation is to introduce Majorana terms softly-breaking D 4 .The resulting mass matrix has now more parameters than observables, and the mixing angles can be fitted in agreement with experiment.
Interestingly, some features remain.For example, realize that in the limit β → 0 the solar and reactor mixing angles go to zero as well as the first family lepton masses.This can be obtained from noting that, in the limit β → 0, Eqs. ( 28) and (29) take the form thus implying that the effective Majorana mass matrix for neutrinos turns into and also favoring the normal ordering case, m ν3 > m ν2 > m ν1 .

VI. DISCUSSION
The main feature of the presented model is that, when tan β → 0, all the quark mixing becomes trivial while in the lepton sector only the solar and reactor mixing angles follow the same fate.The atmospheric one is the only non-zero mixing angle.However, it does receive contributions that could account for 15% of its experimental value.Furthermore, in the same limit, the complete first generation of fermions become massless.Hence, all mixing angles are proportional, or at least related, to tan β.This is the main result of our discussion.Now, it is important to question if this result could have been achieved with other symmetry groups.To investigate it we need to realize that Eq. ( 1) is a direct of the mass matrices having the form up to some possible minus signs; this kind of matrix is then diagonalised by Eq. ( 26).This structure points to those symmetry groups which have doublets in their irreducible representations.We are mainly interested in those types of groups where the tensorial product of two doublets contains: i) at least one doublet with two singlets or ii) no doublets but four singlets.The latter are characterised by D 2n , Q n , Σ n , etc. [7].In the case of the former possibility, to generate Eq. ( 33) we still require an additional auxiliary symmetry, e.g.Z k , to forbid the extra terms 1 .Notice that among all the different choices, the most minimal is the one realized here with D 4 .A detailed study of the phenomenology of this model is left for future work.

VII. CONCLUSIONS
We have considered a generalization of the original idea given in Ref. [6] where the Cabibbo angle was expressed in terms of the physical parameter β commonly appearing in 2HDMs, θ c = 2β.The original proposal only had the two heavy quark generations with non-zero masses and no mixing allowed with the third generation.By adding a third Higgs doublet with a small VEV and assuming a conveniently large decoupling mass, we have been able to sufficiently perturb the original model and not only write all the quark mixing angles in terms of β also do the same in the lepton sector.However, only for the latter case, it was necessary to softly-break the flavor symmetry with generic Majorana terms.Additionally, we have discussed how the suggested relation between these two angles (θ c and β) is a consequence of symmetry groups with doublets in their irreducible representations satisfying tensorial products of the type: i) , where for the former case an auxiliary symmetry is still required to forbid extra terms not appearing in the latter case.Finally, although this framework contains tree-level FCNCs they are sufficiently suppressed as already shown in Ref. [6] and the addition of a very heavy third Higgs doublet represents no significant changes to this picture.
1 Take for example S 3 .