Radiatively generated hierarchy of lepton and quark masses

We propose a model for radiatively generating the hierarchy of the Standard Model (SM) fermion masses: tree-level top quark mass; 1-loop bottom, charm, tau and muon masses; 2-loop masses for the light up, down and strange quarks as well as for the electron; and 4-loop masses for the light active neutrinos. Our model is based on a softly-broken $S_{3}\times Z_{2}$ discrete symmetry. Its scalar sector consists only of one SM Higgs doublet and three electrically neutral SM-singlet scalars. We do not need to invoke either electrically charged scalar fields, or an extra $SU_{2L}$ scalar doublet, or the spontaneous breaking of the discrete group, which are typical for other radiative models in the literature. The model features a viable scalar dark matter candidate.

Here we propose an economical radiative model, which explains the fermion mass hierarchy by a sequential loop suppression, so that the masses are generated according to: t-quark → tree-level mass from q jL φu 3R , their entries are generated at different loop-levels: where j, k = 1, 2, 3. Since the SM fermion masses appear after the electroweak symmetry breaking, the mass matrices are proportional to the VEV v = φ 0 of the SM Higgs, φ, or v 2 in the case of the neutrinos. The latter is the generic consequence of the fact that with the SM-fermion content, the only possibility for the neutrino mass is the Majorana option, described by the Weinberg operator LL c φφ. The mass parameter Λ in Eq. (1.8) is the scale of this operator, which will be introduced in what follows.

The model setup
First we specify our model, allowing for the implementation of the above described setup and then discuss the justification of its structure. Let us stress right from the beginning that we do not pretend to explain the fine details of the mass matrices (1.5)-(1.8) and therefore nor to predict the experimental values of the quark and lepton masses and mixings.
Our goal is more moderate: to provide a mechanism underlying the hierarchy (1.9)-(1.12).
Towards this end we extend the SM gauge group G SM = SU 3c × SU 2L × U 1Y with the discrete symmetry factor which is selected to be the least extra symmetry necessary for the suppression pattern in Eqs. (1.1)-(1.4). More comments will be given below. The field content of the model consists of the SM fields augmented only with SU 2L singlets.
The scalar sector consists of the SM doublet Higgs φ and three SM-singlets σ 1 , σ 2 , η, with the S 3 ⊗ Z 2 assignments: Let us note that aforementioned scalar content is the minimal required to implement the radiative mechanism of the SM fermion mass hierarchy generation (1.
Here 1 and 1 are the trivial and nontrivial S 3 singlets, respectively.
With this field content, the relevant quark, charged lepton and neutrino Yukawa terms invariant under the symmetry (2.1) take the form: We want that after the spontaneous breaking of the electroweak symmetry the abovegiven Yukawa interactions generate the SM fermion masses according to (1.9)-(1.12). This happens if we introduce soft Z 2 breaking terms in the sector of the electroweak singlet fermions as well as soft S 3 breaking in the electroweak singlet scalar sector for the S 3 scalar doublet σ = (σ 1 , σ 2 ). From the interactions (2.4)-(2.7) there emerge 1-, Fig. 1. They implement the loop hierarchical pattern of the SM fermion mass matrix entries (1.10)-(1.12). The top-quark entry κ j3 , according to the field assignments in Eqs. (2.3), is generated at tree-level as declared in (1.9).
Let us comment on the model setup (2.1)-(2.3). In its elaboration we were guided by minimality arguments, compatible with the hierarchical structure (1.9)-(1.12). The selection of the discrete group (2.3) is motivated by the following reasons. The S 3 is the smallest non-abelian group having a doublet irreducible representation, necessary to set up a minimally non-trivial structure of Yukawa interactions (2.4)-(2.7), leading to (1.9)-(1.12). We also need a preserved Z 2 symmetry to separate the exotic F

Model Phenomenology
Let us recall that our goal is to explain the hierarchy (1.9)-(1.12), without pretending to predict the values of the quark and lepton masses and mixings. Nevertheless, resorting to reasonable assumptions about the model parameters and using (2.12), we are able to give several predictions at least at an order of magnitude accuracy.
In the neutrino sector, from the 4-loop diagrams in Fig. 1, it follows that the light active neutrino mass matrix takes the form where m 1 and m 2 are the heavy right handed Majorana neutrino, ν sR , masses and f Assuming y (ν) 2 · f (ν) ∼ 1, µ η ∼ µ σ ∼ m s ∼ α · Λ and taking Λ = 2.5TeV from the quark sector (2.12) we find for α ∼ 1 the light neutrino mass scale m ν ∼ 1eV, which is too heavy. However in our model all the particles are lighter than the cutoff scale Λ. Assuming, for instance, α = 0.3 we arrive at the correct neutrino mass scale m ν ∼ 50 meV.
Let us survey the possible dark matter (DM) particle candidates in our model. Due to the preserved S 3 × Z 2 symmetry, this role could be assigned either to the right handed Majorana neutrinos ν sR or to the lightest of the scalar fields η, Re(σ s ) and Im(σ s ) (s = 1, 2). Here we analyse the case in which the SM singlet Z 2 -odd scalar particle η is lighter than the σ 1 and σ 2 scalars. In this mass range the η is stable. In fact, the only possible decay modes of η are with s, k = 1, 2 and i = 1, 2, 3. These decays may arise from the first and second terms of the charged fermion Yukawa interactions of Eqs. (2.4), (2.5) and (2.6) as well as from the first term of the neutrino Yukawa interaction of Eq. (2.7), respectively. For η lighter than σ 1,2 , the decays (3.3) are kinematically forbidden, and as a result, the η is stable, as necessary for a DM particle candidate. Let us estimate its relic density according to (c.f. Ref. [16]) where σv is the thermally averaged annihilation cross-section, A is the total annihilation rate per unit volume at temperature T and n eq is the equilibrium value of the particle density, which are given by [17] with K 1 and K 2 being the modified Bessel functions of the second kind order 1 and 2, respectively [17]. For the relic density calculation, we take T = m η /20 as in Ref. [17], which corresponds to a typical freeze-out temperature. We assume that our DM candidate η annihilates mainly into W W , ZZ, tt, bb and hh, with annihilation cross sections [18]: where √ s is the centre-of-mass energy, N c = 3 is the color factor, m h = 125.7 GeV and Γ h = 4.1 MeV are the SM Higgs boson h mass and its total decay width, respectively. respectively. The horizontal line shows the observed value Ωh 2 = 0.1198 [19] for the relic density.
value Ωh 2 = 0.1198. As can be seen, the Relic density is an increasing function of m η and a decreasing function of λ h 2 η 2 . In our model we expect a typical mass scale for all the non-SM particles -the η-DM candidate, in particular, -to be m non−SM ∼ m η ∼ α · Λ ∼ 750 GeV. This is hinted, as motivated below Eq. (3.2), by the light neutrino mass scale m ν ∼ 50 meV.
For this DM particle mass, as seen from Fig. 2, the quartic coupling is λ h 2 η 2 /(4π) ≤ 1, which corresponds to the perturbative regime of the model.
Here we suppressed all the super-and subscripts of the fields, unessential for our discussion.
We to this group. Thus we find the S 3 ⊗ Z 2 assignments for these heavy scalars: where ξ i , Φ i (i = 1, 2) and ϕ are SU 2L doublets with hypercharge of 1 2 (as the SM Higgs doublet φ), whereas χ i (i = 1, 2) and ρ are SM singlets with zero hypercharge.
In this particular UV completion the non-renormalizable operators (4.1), (4.2) are replaced with the renormalizable interactions corresponding to the vertices of the diagrams in Fig. 3.
The Yukawa and scalar self-interaction couplings we denoted with z and µ, λ, respectively. This is just one of many possible UV completions of our effective model. In the present paper we do not intend to list all of them, although this is a quite straightforward group theory exercise. Going upwards in the energy scale with a particular UV model one may speculate on the unification to an extended gauge symmetry group relating some parameters of the Lagrangian (2.4)-(2.7) and making the framework more predictive. This extended symmetry should be spontaneously broken down to the group (2.1) at a scale above Λ. This study is beyond the scope of the present paper.

Conclusions.
We have proposed the first model with the SM fermion mass hierarchy generated by the loops. We constructed a model setup as the minimal extension of the SM which allowed us to realize the radiative mechanism. The model does not pretend to explain the quark and lepton masses and mixing angles, but only the mass hierarchy. Nevertheless, through reasonable assumptions about the model parameters we were able to make several rough predictions with order of magnitude accuracy. The model contains non-renormalizable operators with a cutoff scale Λ, which separates the dynamic particles with masses ≤ Λ from the heavy frozen degrees of freedom. We estimated this scale to be Λ ∼ 2.5 TeV, from the 1-and 2-loop quark masses (2.10), (2.11). In the neutrino sector our model predicts -independently of the model parameter values -one massless and two non-zero mass neutrinos: a mass spectrum compatible with the neutrino oscillation data. Analyzing the 4-loop neutrino mass (3.2), we hinted that the mass scale of the non-SM particles of our model are of the order of 1 TeV. Due to the discrete symmetries, our model possesses DM particle candidates. We found that one of them, the SM-singlet scalar η lighter than the other non-SM scalars, could be a viable DM particle. We also commented on the possible implications of the exotic colored fermions for LHC searches. Finally we discussed one of the ultraviolet completions of our effective model.