Sequentially loop supressed fermion masses from a unique discrete symmetry

We propose a systematic and renormalizable sequential loop suppression mechanism to generate the hierarchy of the Standard Model fermion masses from one discrete symmetry. The discrete symmetry is sequentially softly broken in order to generate one-loop level masses for the bottom, charm, tau and muon leptons and two-loop level masses for the lightest Standard Model charged fermions. The tiny masses for the light active neutrinos are produced from radiative type-I seesaw mechanism, where the Dirac mass terms are effectively generated at two-loop level.


Introduction
Despite its striking consistency with experimental data, the Standard Model (SM) does not explain several fundamental issues, such as the number of fermion generations, the observed pattern of fermion masses and mixings, etc. In fact, the huge fermion mass hierarchy spans over a range of 13 orders of magnitude from the light active neutrino mass scale up to the top quark mass.
In the SM fermion masses arise from a Dirac mass term, which is completely invariant under the SM gauge group. Take for instance the Dirac term for the u-type quarks, HereH = iσ 2 H * , q Lα is the left-handed quark doublet and u Rβ the right-handed u-quark singlet. As the three quark generations (α, β = 1, 2, 3) transform equally under the SM, all three masses will be generated by the same tree level Yukawa matrix after electroweak symmetry breaking (EWSB) and they should naturally have a similar value. However, in actual practice this is not the case as quark and lepton masses span several orders of magnitude. Thus, in order to accommodate the experimental values of the charged fermion masses in the SM, one needs to rely on an unnatural tuning among their different Yukawa couplings. All these unexplained issues strongly indicate that new physics has to be invoked to address the fermion puzzle of the SM. To tackle the limitations of the SM, various extensions, including enlarged scalar and/or fermion sector, as well as extended symmetries, discrete or continuous, have been proposed in the literature .
A possible scenario to address the SM flavor puzzle is to effectively generate the masses of the different fermion generations via other mechanisms, such as radiative mass generation, in which different fermions have distinct loop order suppression. It is natural to start with the top mass generated at tree level, then the charm and bottom masses should appear at one-loop level, while the up, down and strange masses arise at two-loop level. For the leptonic sector, the muon and tau masses are generated at one-loop level, the electron mass at two-loop level, while the light active neutrinos have to acquire mass, at least, at four-loop level if they are of Dirac type, although this can be relaxed considering Majorana neutrinos with a seesaw-like mechanism [81]. For this realization to be possible, one needs an extra symmetry beyond the SM to differentiate between generations. This mechanism was proposed for the first time in [83] in a non-renormalizable approach, while renormalizable versions have been implemented in [84,85]. However, such renormalizable models rely on rather large scalar and fermion sectors, sophisticated symmetries and lack a guiding principle for the model construction.
In this work we propose a systematic and renormalizable sequential loop suppression mechanism to generate the hierarchy of fermion masses from a unique softly broken discrete Z n symmetry. The symmetry and the charges of the fields under it are chosen in such a way that each loop level breaks the discrete symmetry, but keeps a remnant exact subgroup of it, which is responsible for protecting the hierarchy.
The paper is organized as follows. In section 2 we describe how to generate a sequential loop suppression, using just one symmetry beyond the Standard Model. This mechanism is then implemented for the quark and leptonic sector using a Z 4 symmetry in sections 3 and 4, respectively. In section 5 we describe a set of possible charge assignments for the fields that mediate the quark and lepton radiative masses given in the previous sections. Finally, we shortly discuss the implications of the soft breaking mass terms of such models in section 6 and present our conclusions in section 7.

Loop suppression mechanism
In this section we describe the general guidelines followed in order to generate a sequential loop suppression, using just one symmetry beyond the Standard Model. This mechanism is to be applied to the dynamical generation of Standard Model fermion masses, mimicking their hierarchy via loop orders. However, this framework goes beyond the application shown here regarding only the SM fermions and can be implemented in other scenarios.
For simplicity, we will use the operator O β ≡q LβH u Rβ in (1) to illustrate how the mechanism works. The generalization for any non-diagonal Dirac flavor mass operator is straightforward. In the SM the operators O 1 , O 2 and O 3 are indistinguishable from a symmetry point of view, and for this reason, a new symmetry is needed to differentiate them. Then this symmetry has to be broken at some scale, allowing for the existence of the operators O i in the low-scale theory, with suppression factors for each of them. The main strategies followed in the literature consist in working with the low energy scale directly, breaking softly the symmetries [83]; or to generate these operators effectively by adding scalars that break spontaneously the new symmetries [84,85] Here we restrict ourselves to the case of (i) renormalizable Dirac Yukawa mass operators like (1) and (ii) a new softly broken symmetry. For simplicity we consider only abelian discrete symmetries. This new symmetry should forbid a Dirac Yukawa mass term at all orders, but once it is broken softly, an effective realization of this term via loops is allowed, while it is still forbidden at tree level, since it is a hard dim=4 term. This same symmetry can also be used to distinguish between fermion generations, allowing the Dirac Yukawa mass operators to be realized at different loop orders for each generation. Nevertheless, a general implementation of this idea is sometimes difficult, if we consider just a unique symmetry beyond the SM, since naive control of its breaking can spoil the desired features of the constructed model. For instance, when breaking softly the symmetry in order to allow O 2 , one must ensure that the same mass mechanism is not allowed for O 1 . Otherwise it would not be possible to introduce the desired hierarchy to the quark mass spectrum.
In this work we show a mechanism to generate effectively a mass hierarchy of operators among fermion generations, by softly breaking a unique symmetry group G, while still protecting the hierarchy. Considering abelian discrete groups, the main requirement is that this extra symmetry group G is not simple, and it contains at least one non-trivial normal subgroup H. Now consider that the two operators O 1 and O 2 transform non-trivially under the new symmetry G as, where {1} is the identity group and h ∈ H −{1} denotes any element of H except the identity. Analogously, g ∈ G − H refers to any element in G but not in H.
In order to realize O 2 at loop level, a soft breaking term transforming as h −1 ∈ H − {1} is included. Because H is a subgroup, no product of elements in H can generate an element in G − H. For this reason, no combination of O 2 with the soft breaking term can generate O 1 at any level. To generate O 1 , a soft term transforming as g −1 ∈ G − H is required. Consequently, the extra symmetry ensures that the operators O 2 and O 1 can be generated through different realizations even after the symmetry has been broken. 1 For instance, providing non-reducible diagrams [86,87], O 2 can be realized at one-loop level, while O 1 can be constructed at a two-loop level.
It is worth mentioning the fact that after breaking the subgroup H with the soft-breaking term the operator O 1 cannot be closed, which means that there is a remnant symmetry protecting O 1 . This residual symmetry is the quotient group G/H.
Note that this reasoning can be applied to as many operators as long as the subsequent subgroups are not simple. For instance, following this idea one can systematically protect the loop realization of n Dirac mass operators Moving to a concrete example to illustrate this mechanism, the simplest groups for which a systematic chain of normal subgroups can be found are the cyclic Z n groups, with n not a prime number. Among them, the easiest of these chains is Z 2 ⊂ Z 4 ⊂ Z 8 ⊂ ... ⊂ Z 2 n . As we want to apply this to the SM, we only need to generate masses up to the two-loop level, so we can take a Z 4 symmetry for simplicity. Considering as before the u-mass operators O β (1), we want to ensure that the mass of the top quark be generated at tree level, while the charm and up quarks arise at one-and two-loops, respectively. Following the arguments given above, for G ≡ Z 4 and H ≡ Z 2 , so that Z 4 /Z 2 ∼ Z 2 , we have: • O 3 ≡q LαH u R3 should transform as a singlet under Z 4 , so the top mass can be realized at tree level.
• On the other hand, O 2 transform as −1 ∈ Z 2 − {1} ⊂ Z 4 and O 1 as ±i ∈ Z 4 − Z 2 ⊂ Z 4 . At the moment, neither of these two operators are allowed by the symmetries.
• A soft-breaking term, transforming as −1 under Z 4 , is added in order to close the mass term for the charm quark O 2 ≡q LαH u R2 at one-loop level.
• This soft breaking term breaks Z 4 down to a remnant Z 2 group. The operator O 1 still cannot be closed since it transforms as −1 under Z 2 .
• A soft-breaking term, transforming as −1 under the remnant Z 2 (or equivalently as ∓i under the original Z 4 ), is then added to realize O 1 ≡q LαH u R1 , breaking completely the initial Z 4 symmetry and generating the mass of the u-quark at two-loop level.
As a result, starting from a simple Z 4 symmetry, one can ensure that the mass mechanisms for the three u-quark families are decoupled, realized at different orders and protected through the sequential breaking of the cyclic Z 4 symmetry.

Quark sector
The idea that discrete symmetries are behind the pattern of the hierarchy of quark masses via loop suppression is not new in the literature [83]. However, in spite of various interesting developments, the previous models require many symmetries and assumptions in order to fix the specific fermionic patterns. In the next sections we illustrate the loop suppression mechanism explained before to generate the SM fermion mass hierarchy, based on a unique Z 4 symmetry. For the quark sector, we consider that the charm and bottom quark masses arise at one-loop level, while the up, down and strange quark masses are generated at two-loop level. On the other side, the top quark mass is generated at tree level from the standard renormalizable Yukawa operator. The diagrams that generate the d-type and u-type quark masses are depicted in Fig. 1 and Fig. 2, respectively. We give explicitly, between parenthesis, the charge of each field under Z 4 , using as representation the 4-th roots of unity (1, i, −1, −i). The rest of the SM quantum numbers are shown in Sec. 5. It is straightforward to see that the vector-like masses (⊗) of the new fermions F break softly the Z 4 symmetry, allowing the loop to be closed. This symmetry is broken softly by the vector-like masses denoted by ⊗.
The role of the Z 4 symmetry and its subgroup Z 2 in the loop suppression mechanism can be directly seen from the diagrams. Take for instance Fig. 2. All the fields in the one-loop diagram, which generate the operator O 2 ≡q LαH u R2 , transform as ±1 under Z 4 , i.e. they all belong to the Z 2 subgroup. Meanwhile, H(1) Figure 2: One-and two-loop diagrams responsible for the masses of the SM up-type quark mass matrix.
Here α = 1, 2, 3. The diagrams are generated once the discrete Z 4 symmetry is softly broken by the fermion masses, denoted by ⊗. The charges under Z 4 are given between parenthesis next to the corresponding field.
the two-loop diagram (O 1 ) contains new fields transforming as ±i, which belong to the subset Z 4 − Z 2 , realizing the mechanism described in Sec. 2.
The mass diagrams are generated by the following Yukawa interactions for the u-type and d-type quark sectors, as well as the terms of scalar potential given by The number of generations considered for the new fermions is the minimal possible in order to reproduce all the SM quark masses. Notice that the first line of (4) generates a tree level top quark mass. The terms on the second and third lines enters in the one-and two-loop Feynman diagrams of Fig. 2, which generate the second and first columns of the u-type quark mass matrix, respectively, thus giving masses to the charm and up quarks. Similarly, for the SM d-type quark mass matrix, the terms on the first and second lines of (5) are associated to the one-and two-loop Feynman diagrams of Fig. 1, respectively, i.e. the different entries of the SM d-type quark mass matrix. This results in a one-loop level bottom quark mass and two-loop down and strange quark masses.
In order to close the loop Feynman diagrams, crucial for the implementation of sequential loop suppression mechanism, the following soft-breaking mass terms for the exotic fermions are required: The terms F 1R F 1L and F 2R F 2L break Z 4 in two units, leaving a Z 2 residual symmetry. These are associated to the one-loop diagrams. The two-loop diagrams break completely Z 4 , as they are generated via the soft breaking term F 3R F 3L , which transforms as i.
In principle, more soft breaking terms may arise depending on the SM quantum numbers of the fields for a specific model. Nevertheless, this is model dependent and cannot be treated in general. This does not represent any problem except for certain particular choices of the internal fields, for which an extra symmetry may be needed. This is the case for the particle content chosen in Sec. 5.

Leptonic sector
Analogously to the quark sector, the same diagrams and Z 4 charges can be applied to generate the lepton masses. Of course, as the quantum number of the external particles are colour blind, one should add either scalar leptoquarks or massive colourless vector-like fermions, as it will be discussed in Sec. 5.
The diagrams depicted in Figs. 3 and 4 for the charged lepton and neutrino masses are generated by the following Yukawa interactions terms, respectively, as well as the relevant scalar interactions, Likewise the quark sector, here in order to close the loop diagrams we require the following soft-breaking mass terms for the charged exotic fermions, Ψ, and right-handed Majorana neutrinos N R . After the electroweak symmetry breaking the above-given Lagrangian (8)-(10), along with the Z 4 softbreaking mass terms (11), (12), lead to one-loop level masses for the tau and muon, and a two-loop mass for the electron, shown in Fig. 3. A two-loop Dirac mass term for neutrinos arises from the Feynman diagram in Fig. 4, analogously to the Dirac mass for the up quark, Fig. 2. In principle, for similar values H(1) Figure 4: Two loop Feynman diagram that generates the Dirac neutrino mass term effectively. Here α = 1, 2, 3 and β = 1, 2. After softly breaking Z 4 the Majorana mass terms of (12) should be included, thus giving rise to an effective seesaw neutrino mass, see Fig. 5.
of the couplings and masses, the neutrino mass would be roughly of the order of the electron mass, as both are generated through a similar mechanism. However, since the initial Z 4 symmetry is completely broken, a Majorana mass term is allowed for the fermions N R in (12). Therefore, one can generate an effective four-loop Majorana mass for the light active neutrinos via the radiative seesaw Type-I mechanism shown in Fig. 5 and first described in [81]. Given the seesaw relation for the lightest neutrinos and since m D is generated at two-loop level by the diagram Fig. 4, the light active neutrino mass scale comes with a factor of ( 1 16π 2 ) 4 , which could guarantee a strong neutrino mass suppression even for a relatively small Majorana mass scale M R . Figure 5: Radiative seesaw Type-I mechanism for neutrinos [81]. The blob denotes the Dirac mass term generated at two-loop level through the diagram in Fig. 4. The symbol ⊗ indicates the Majorana mass for N R softly-breaking the Z 4 . Generation indices are suppressed for simplicity.
The Dirac mass can be computed straightforwardly from the diagram of Fig. 4 and the neutrino Yukawa terms given in (9), with the result: where v ≡ H 0 and I loop is a dimensionless two-loop function.
Here we are only interested in estimating the characteristic neutrino mass scale of this model. For this reason, we consider the unrealistic case with one massive neutrino, by assuming no hierarchy or flavor structure in the Yukawa couplings and just one copy of the new fermions, i.e. y (ν) Moreover, as the masses of the Ψ 2 , Ψ 3 and N R exotic fermionic fields are associated with the softly breaking scale of Z 4 , we consider, for the sake of simplicity, a scenario where they are diagonal and degenerate, i.e., M = M R , and also for the scalar fields mediating the radiative processes, m Φ5 = m Φ6 = m S . Then, the Dirac neutrino mass scale given by (14) can be estimated as, where now the loop integral is a dimensionless function of just the ratio of the scalar/fermion masses given by This integral can be decomposed in terms of one master integral, for which an analytical solution can be easily found [27,88].
The mass scale of the lightest active neutrino can be directly estimated through the seesaw relation shown in (13), yielding the following estimate, This mass scale as a function of M R is plotted in Fig. 6 for several scalar masses. We can observe that the difference, compared with the standard seesaw type-I/III, arises from the fact that the neutrino mass m ν is considerably suppressed even for small values of the Majorana mass M R , due to the effective four-loop level suppression. For example, if we set the values of the couplings to O(1), one can fit the atmospheric neutrino oscillation scale m ν ∼ 0.05 eV [89] with the relatively small Majorana mass of the order of 100 TeV. This should be compared to the typical Majorana mass scale of 10 14 GeV in the standard seesaw mechanism with order 1 couplings. It is worth mentioning that taking into account the limits from big bang nucleosynthesis [90] and ∆N eff [91], which set a rough lower bound of M R (0.1 − 1) GeV, this class of models are severely constrained [81]. From Fig. 6 we can see that even for the scalar masses of order 1 TeV and couplings O(1), in order to get a reasonable neutrino mass without large non-perturbative values of the couplings in (17), the values of M R should lie between 1 GeV and 10 7 GeV.
Experimental bounds from the searches for lepton number violation are applicable for the range 100 MeV ≤ M R ≤ 500 GeV [92], depending on the value m S . Also limits on the Yukawa couplings and M R are available from the experimental searches for Lepton flavor violating processes, like µ → eγ [93]. However, our goal here is to provide an order-of-magnitude estimate of the characteristic neutrino mass scale for this model without going into details. A more insightful study, including fitting neutrino oscillation data, should be done in order to set well-grounded limits to this type of neutrino mass models.

Particle content
In this section, we describe the charge assignments for all the fields mediating the previously discussed radiative processes, which give masses to the SM quarks lighter than the top quark and to the SM leptons. We discuss different scenarios, characterized by the specific nature of the internal fields mediating the loop. Table 1 gives the SM and N R charge assignments under the Z 4 discrete group. These fields are common to all the models. In Tables 2, 3 and 4, the extra particle content used in each model on top of the SM fields is described. In Comments, the possible nature of the fields inside the loop is given, in order to provide an insight to the phenomenology of these particles. Depending on the SU (3) and SU (2) charge assignments, the new fermions F L/R and Ψ L/R can be considered either as vector-like colored fields or vector-like non-colored fields. At the same time, the scalar sector could be colored or non-colored too. All the quantum numbers are common for all the generations of a given field, except for the Z 4 charges, which are generation dependent.
As described in the previous sections, in order to generate the operators, which give masses to the quark and leptons radiatively, we should softly break the discrete symmetry inside the loops, in our case Z 4 or the remnant Z 2 . Once the symmetry of the model is broken, for consistency we should add to the theory all the possible soft breaking terms. Some of these terms, combined with the interactions of the SM fields, the extra fermions and the scalars inside the loop, may be dangerous in the sense that they could allow the SM quarks and leptons to acquire masses at tree level. Therefore, if we want the loop diagrams to be leading contributions to the fermion masses, we must forbid the tree level mass operators by an extra symmetry such as a global U (1) X . Note that the necessity of this U (1) X is a model dependent feature due to our choice of the SM quantum numbers of the particles in the loop and it is not related to the the above-described loop suppression mechanism based on a discrete symmetry, which is model independent.
(1, 2, 1/2) 1 Table 1: Charge assignments for the SM fields and the right-handed field N R , common to all the models. Not all the fields are charged under the extra global U (1) X symmetry. We consider the standard number of generations α = 1, 2, 3, while β = 1, 2 , to give masses to two active light neutrinos, which is the minimal phenomenologically viable choice. In the last column in the brackets we indicate Z 4 charges for the three generations.

Option 1: No colored scalars
Let the new scalars be SU (3) C singlets, so the same set of scalars can, in principle, mediate both quark and lepton masses. In this case, specified in Table 2, the number of scalars is considerably reduced in comparison to the case displayed in Figs 1 -4. Here, the scalars mediating the quark masses, S i , can be identified with those participating in the generation of the lepton masses, i.e. S i ≡ Φ i . Moreover, we may set S 6 = S 5 (Φ 6 = Φ 5 ) and Ξ = ∆ = S 2 , further simplifying the scalar content. On the other side, this scenario requires vector-like colored and non-colored fermion fields, participating separately in the quark and lepton masses. It is worth mentioning that this scenario will have a rich phenomenology, containing vector-like copies of all the SM fermions, for which there is an extensive literature on experimental searches and phenomenological studies [94][95][96][97][98]

Option 2: All vector-like fields are colored
Here we consider the possibility that all the vector-like fermion fields carry a non-trivial SU (3) color ch n this case, the same specified in Table 3 fermions that mediate the quark masses can also enter the lepton masses, i.e. Ψ i ≡ F i . In contrast to the previous example, here one needs to introduce color charged scalars in order to generate lepton masses. For this case also, Ξ = S 2 .
In this scenario, as the quark and lepton masses are generated from the same soft breaking terms, a correlation between both sectors may be found for simplified cases. The phenomenology of vector-like quarks Table 2: Charge assignments for the fermions and scalars contributing to the diagrams Figs. 1-4, considering that all the scalars are color blind. In the brackets of the Z 4 -column we indicate the corresponding charges of the left-handed and right-handed vector-like fermions, respectively. The U (1) X global symmetry and the field charges are discussed in section 6.
Fields Table 3: Charge assignments for the fermions and scalars that participate in the diagrams Figs. 1-4 considering that the same set of colored fermions mediate quark and lepton masses, i.e. Ψ i ≡ F i . Note that in this case colored scalars with U (1) X charges appear. Other notations are the same as in Table 2. apply to this model, giving a rough conservative limit of O(1) TeV to their masses [96,97]. Furthermore, limits on the scalar leptoquark masses can be found in Refs. [99][100][101][102].

Option 3: All vector-like fields are color singlets
We finally let us analyze the possibility that none of the vector-like fields carry color charge. In this case specified in Table 4, the same fermions that mediate the lepton masses can also enter the quark masses, i.e. F i ≡ Ψ i . Leptoquark scalars are also needed. The scalar sector can be simplified by setting ∆ = Φ 2 and Φ 6 = Φ 5 . Likewise the previous case, correlations between the lepton and quark masses can sometimes be found. For the limits on the vector-like lepton masses we refer to [94,95].  Table 2.
All the examples of models discussed here represent just a small set of the possibilities. We have restricted ourselves to the models containing fields up to the SU (3) C -triplets and the SU (2) L -doublets with hypercharge below two. Other particle contents with both colored and non-colored vector-like fermions can be of interest too, as well as more exotic fields, for which a rich phenomenology may exist. It is worth mentioning that a scalar leptoquark with the SM quantum numbers (3, 1, -1/3) can give rise to proton decay at tree level [103,104]. Usually, proton decay is forbidden by imposing either a discrete cyclic symmetry [105], or a U (1) symmetry, which can be either local [104] or global. As discussed in the next section, a global U (1) X symmetry is implemented in our framework. Under this symmetry the SM fermionic fields are neutral, whereas the scalar leptoquarks are charged. As a result, the tree-level operators that induce proton decay are forbidden.

About the soft breaking
In this section we explain the structure of the soft breaking terms, motivating the introduction of the global U (1) X symmetry in our models. Given a set of symmetries and fields, in order to be sure that the theory is renormalizable, one should include all the terms allowed by the symmetry with mass dimensions up to four. When some symmetry is softly broken, i.e. by introducing explicit breaking terms with dimension-3 or below (soft terms) in the interaction Lagrangian, no divergences appear for dimension-4 interaction terms (hard terms). However, divergencies may arise, if one does not add all the soft terms that break the symmetry in the same way, therefore, spoiling renormalizability. A conservative approach, to be sure that the theory is still renormalizable after soft-breaking, is to include in the Lagrangian every soft term allowed by the remnant symmetries.
In all the radiative models described above, the Z 4 symmetry is broken by adding soft terms depicted as ⊗ in the diagrams. In principle, as the symmetry is broken, we have to write every soft term invariant under the SM. Considering, for example, the one-loop diagram describing the bottom quark mass Fig. 1 (left) with the field content of Table 2, Z 4 is broken to a Z 2 by the soft termF 1R F 1L . This implies that the soft term HS 2 has also to be included in the Lagrangian in order to absorb infinite contributions arising from diagrams like Fig. 7.
Therefore, S 2 will mix with the SM Higgs boson H and generate a tree-level mass for quarks through the termF 1R q α S 2 . We add the U (1) X symmetry to protect the models from these issues, requiring that it Figure 7: Two-loop diagram that generates an infinite contribution to the soft term HS 2 . Note that this diagram is built directly from the one-loop bottom quark mass Fig. 1 (left) by closing the fermion lines. This diagram implies that once the soft termF 1R F 1L (⊗) is added, HS 2 should be included too. If not the resulting theory would not be renormalizable.
forbids the mixing of the internal loop particles with the SM ones. The condition that the vertices given in Figs. 2-4 are allowed, while the undesired soft terms are not, defines a system of linear equations for the U (1) X charges. Its solution, considering that the SM fields are not charged, is given in Tables 2-4 in terms of a free parameter x, with x = 0.
It is worth mentioning that the existence of soft terms as HS 2 is completely model dependent. In our case, it is due to our choice of the quantum numbers of S 2 in section 5.1, which transforms as the Higgs under the SM. For other choice of the quantum numbers of the internal fields, such couplings may not exist and one can decouple the internal particles from the SM without, in principle, any new symmetry. Nevertheless, the same U (1) X global symmetry we explicitly included here will still be a symmetry of the theory but at the accidental level.

Discussion
We proposed a systematic method of construction of renormalizable models of the SM fermion masses with the observed hierarchy originating from the sequential loop suppression mechanism resorting to a unique softly broken discrete symmetry. We applied this method to an example model based on Z 4 -symmetry. In this model non-SM fields and the softly broken Z 4 symmetry are devised to generate 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 effective 4-loop masses for the light active neutrinos. The 4-loop level masses for the light active neutrinos arise from the radiative Type-I seesaw mechanism, where the Dirac mass terms are generated at two loop level. To illustrate the sequential loop suppression mechanism we considered three scenarios: 1) With no colored scalars. 2) With all vector-like fields colored. 3) With no colored vector-like fields. We also discussed self-consistent inclusion of the soft breaking terms needed for the implementation of the proposed mechanism of the sequential loop suppression in the way that undesirable radiatively induced soft terms are suppressed by an accidental U (1) X symmetry.
Due to this accidental symmetry, the models considered in this paper have several stable scalar and fermionic dark matter candidates. Considering a scenario with a scalar DM candidate, one has to ensure its stability, which can be done by assuming that it is the lightest among the inert scalar particles and is lighter than the exotic fermions. That scalar DM candidate annihilates mainly into W W , ZZ, tt, bb and hh via a Higgs portal scalar interaction where H is the SM Higgs doublet and Φ DM the scalar DM candidate in our model. These annihilation channels will contribute to the DM relic density, which can be accommodated for the appropriate values of the scalar DM mass and of the quartic scalar coupling λ DM , similarly to Refs. [54,55,83,106]. 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 coupling λ DM Given the large number of parameters in the scalar potential of the models considered in this work, there is a lot of parametric freedom that allows us to successfully reproduce the current experimental value of the DM relic density. The parameter space of our model consistent with DM constraints will be similar to the one in Refs. [54,55,83,106].
Finally, it is worth mentioning that the charged exotic fermions in our model can decay into the SM charged fermion and a heavy scalar, which in several cases can be electrically neutral. For that scenario, the heavy electrically neutral scalar can be identified as missing energy. In addition, the heavy charged exotic fermions can be pair produced at the LHC via Drell-Yan annihilation only, if they are leptons, and via gluon fusion and Drell-Yan mechanism, if they are quarks. Consequently, the observation of an excess of events with respect to the SM background in the dijet and opposite sign dileptons final states at the LHC, can be treated as a signal supporting the models considered in this work. The precise signature of the decays of the exotic fermions depends on details of the spectrum and other parameters of the models considered here. It is worth mentioning that models with vector-like fermions are being tested at the LHC, and the ATLAS collaboration has reported several analyses in this direction, in particular [96][97][98]. Currently, the bounds M B > 1. 22 TeV and M T > 1.31 TeV for the exotic isosinglet quark masses have been obtained and presented in Ref. [98], under the assumption that the exotic quarks can decay to the SM particles. Specifically, in those studies it is assumed that the vector-like quarks would be first produced at collider experiments via pair-production, a process dominated by gluon fusion mechanism, gg →BB (T T ). Then, each exotic quark would decay to: T → W b, Zt, Ht or B → W t, Zb, Hb, where only the third fermion family has been considered.
A detailed study of the collider phenomenology of the proposed models in is beyond the scope of this paper and is left for future studies.