Fermion spectrum and $g-2$ anomalies in a low scale 3-3-1 model

Abstract

We propose a renormalizable theory based on the $SU(3)_C\times SU(3)_L\times U(1)_X$ gauge symmetry, supplemented by the spontaneously broken $U(1)_{L_g}$ global lepton number symmetry and the $S_3 \times Z_2$ discrete group, which successfully describes the observed SM fermion mass and mixing hierarchy. In our model the top and exotic quarks get tree level masses, whereas the bottom, charm and strange quarks as well as the tau and muon leptons obtain their masses from a tree level Universal seesaw mechanism thanks to their mixing with charged exotic vector like fermions. The masses for the first generation SM charged fermions are generated from a radiative seesaw mechanism at one loop level. The light active neutrino masses are produced from a loop level radiative seesaw mechanism. Our model successfully accommodates the experimental values for electron and muon anomalous magnetic dipole moments.

1 Introduction

Despite of the excellent agreement of the Standard Model (SM) predictions with the experimental data, there are several problems that do not find explanation within its framework. Among them are the observed pattern of SM fermion masses and mixing angles, the tiny values of the light active neutrino masses, the number of SM fermion families, the electric charge quantization and the anomalous magnetic moments of the muon and electron. Addressing these issues requires to consider extensions of the SM with enlarged particle content and symmetries. In particular, theories based on the $SU(3)_C\times SU(3)_L\times U(1)_X$ gauge symmetry (3-3-1 models) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52] , have received a lot of attention since they answer some of the open questions of the SM, such as, for example, the number of SM fermion families and the electric charge quantization. Adding discrete symmetries and extending the scalar and fermionic content of such 3-3-1 models allows addressing the observed SM fermion mass and mixing hierarchy. Furthermore, if one considers 3-3-1 models where the fermions do not have exotic electric charges, the third component of the $SU(3)_L$ leptonic triplet will be electrically neutral. This allows the implementation of a low scale linear or inverse seesaw mechanism producing the tiny light active neutrino masses and sterile neutrinos with masses at the $SU(3)_L\times U(1)_X$ symmetry breaking scale, thus making the model testable at colliders.

Imposing discrete symmetries allows one to forbid tree level masses arising from the Standard Yukawa interactions for the SM fermions lighter than the top quark. To generate such masses, one has to consider heavy vector-like fermions, mixing with the SM fermions lighter than the top quark, as well as gauge singlet scalar fields. Their inclusion in the particle spectrum of the model is crucial for the implementation of the Universal and radiative seesaw mechanisms needed to generate the masses for the SM fermions lighter than the top quark, thus explaining the SM charged fermion mass hierarchy. In addition, the heavy vector like leptons can provide an explanation for the anomalous electron and muon magnetic moments, which is not given within the context of the SM. A study of such $g-2$ anomalies in terms of New Physics and a possible UV complete explanation via vector-like leptons is performed in [53]. Also in Ref. [54], it was shown that the $g-2$ anomalies can be explained using a minimal supersymmetric SM assuming a minimal flavor violation in the lepton sector. Theories involving extended scalar sector [53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70] as well as vector like leptons [71], heavy $Z^{\prime }$ gauge bosons [51, 72], and conformal extended technicolour [73] have been proposed to explain the $g-2$ anomalies. In this work we will consider a renormalizable theory based on the $SU(3)_C\times SU(3)_L\times U(1)_X$ gauge symmetry, supplemented by the spontaneously broken $U(1)_{L_g}$ global lepton number symmetry and the $S_3 \times Z_2$ discrete group. We choose $S_3$ symmetry since it is the smallest non-Abelian discrete symmetry group having three irreducible representations (irreps), explicitly, two singlets and one doublet irreps. This symmetry has been shown to be useful in several extensions of the SM, for obtaining predictive SM fermion mass matrix textures that successfully describe the observed SM fermion mass and mixing pattern [12, 19,20,21, 74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103] . In the proposed model, the top and exotic quarks get tree level masses whereas the masses of the bottom, charm and strange quarks as well as the tau and muon charged lepton masses are produced from a tree level Universal Seesaw mechanism. The masses for the first generation SM charged fermions are generated from a one loop level radiative seesaw mechanism mediated by charged vector like fermions and electrically neutral scalars. The light active neutrino masses are produced from a one loop level radiative seesaw mechanism. Unlike the 3-3-1 models of Refs. [19,20,21, 31, 32, 37, 42, 45, 50] , where non renormalizable Yukawa interactions are employed for the implementation of a Froggat Nielsen mechanism to produce the current SM fermion mass and mixing pattern, after the discrete symmetries are spontaneously broken, our proposed model is a fully renormalizable theory with minimal particle content and symmetries where tree level Universal and a one-loop level radiative seesaw mechanisms are combined to explain the observed hierarchy of SM fermion masses and fermionic mixing parameters. Furthermore, unlike Refs. [19,20,21, 31, 32, 37, 42, 45, 50] our current work has an explanation for the electron and anomalous magnetic moments. In our current model, the charged vector-like leptons which mediate the tree level Universal and one-loop level radiative seesaw mechanism that generates the SM charged fermion mass hierarchy, make contributions to the measured values of the muon and electron anomalous magnetic moments, thus providing a connection of the fermion mass generation mechanism and the $g-2$ anomalies, which is not given the models of Refs. [19,20,21, 31, 32, 37, 42, 45, 50] . Our model is consistent with the low energy SM fermion flavor data and successfully accommodates the experimental values of the muon and electron magnetic dipole moments.

The content of this paper goes as follows. The model is introduced in Sect. 2. The model predictions for the muon and electron anomalous magnetic moments are discussed in Sect. 3. Section 4 is dedicated to the quark masses and mixings. Lepton masses and mixings are analyzed within the model in Sect. 5. The generation of neutrino masses is discussed in Sect. 6 . Conclusions are given in Sect. 7.

2 The model

We consider a renormalizable extension of the 3-3-1 model with right handed Majorana neutrinos, where the $SU(3)_C\times SU(3)_L\times U(1)_X$ gauge symmetry is supplemented by the spontaneously broken $U(1)_{L_g}$ global lepton number symmetry and the $S_3 \times Z_2$ discrete group, the scalar sector is enlarged by the inclusion of several gauge singlet scalars and the fermion sector is minimally augmented by the introduction of heavy electrically charged vector like fermions. Such electrically charged vector like fermions are assumed to be singlets under the $SU(3)_L$ gauge symmetry, thus allowing to easily comply with collider constraints as well as with the constraints arising from electroweak precision tests. The left and right handed components of such vector like fermions have the same transformation properties under the different group factors of the model thus allowing to build mass terms for these fields invariant under the $SU(3)_C\times SU(3)_L\times U(1)_X\times S_3 \times Z_2$ group. The scalar and fermionic content with their assignments under the $SU(3)_C\times SU(3)_L\times U(1)_X\times S_3 \times Z_2$ group are shown in Tables 1 and 2 , respectively. The dimensions of the $SU(3)_C$, $SU(3)_L$ and $S_3$ representations shown in Tables 1 and 2 are specified by the numbers in boldface. It is worth mentioning that the set of vector like fermions ${\tilde{T}}_{n}$ ($n=1,2$), $B_i$ and $E_i$ ($i=1,2,3$) is the minimum amount of exotic fermions required to generate the tree level masses via Universal seesaw mechanism for the bottom, charm and strange quarks as well as the tau and muon as well as one loop level masses for the first generation SM charged fermions, i.e., the up, down quarks and the electron. To implement such tree level Universal and radiative seesaw mechanisms we have introduced the gauge singlet scalars $\xi _n$ ($n=1,2)$ and $\varphi$. In addition, the remaining gauge singlet scalars $\sigma _i$ ($i=1,2,3$) are crucial to generate the Majorana mass terms necessary to radiatively produce the light active neutrino masses. The vector like fermions mix with the SM charged fermions lighter than the top quark thus giving rise to a tree level Universal seesaw mechanism that produces the masses for the bottom, charm and strange quarks as well as the tau and muon charged lepton masses. The first generation SM charged fermions, i.e., the up, down quarks and the electron get their masses from a one loop level radiative seesaw mechanism mediated by charged vector like fermions and electrically neutral scalars. In addition, light active neutrino masses are generated from a one loop level radiative seesaw mechanism mediated by the right handed Majorana neutrinos and the electrically neutral components of the $SU(3)_L$ scalar triplet $\chi$. The smallness of the light active neutrino masses is attributed to a small mass splitting of the $\chi _{1R}$ and $\chi _{1I}$ scalar fields, which originates from the trilinear term $A\left( \chi ^{\dagger }\eta \sigma _{3}+h.c\right)$ of the scalar potential given in Appendix B. Thus, the trilinear coupling A has to be sufficiently small to provide a natural explanation for the tiny masses of the light active neutrinos. In Sect. 6 we discuss a symmetry-based condition for technically natural smallness of the parameter A. Notice that the $U(1)_{L_g}$ global lepton number symmetry is spontaneously broken down to a residual discrete $Z_2 ^{(L_g)}$ by the vacuum expectation value (VEV) of the $U(1)_{L_g}$ charged gauge-singlet scalars $\sigma _{i}$ ($i=1,2,3$) having a nontrivial $U(1)_{L_g}$ charge, as indicated by Table 1. The residual discrete $Z_2 ^{(L_g)}$ lepton number symmetry, under which the leptons are charged and the other particles are neutral, forbids interactions having an odd number of leptons, thus preventing proton decay. The massless Goldstone boson, i.e., the Majoron, arising after the spontaneous breaking of the $U(1)_{L_g}$ symmetry, does not cause problems in the model because it is a $SU(3)_L$ scalar singlet.

In addition, our model does not have fermions with exotic electric charges. Thus, the electric charge in our model is defined as follows:

$$\begin{aligned} Q=T_3+\beta T_8+X=T_3 -\frac{1}{\sqrt{3}}T_8+X\, . \end{aligned}$$

(1)

Furthermore, the lepton number has a gauge component as well as a complementary global one, as indicated by the following relation:

$$\begin{aligned} L=\frac{4}{\sqrt{3}}T_{8}+L_g, \end{aligned}$$

(2)

being $L_g$ a conserved charge associated with the $U(1)_{L_g}$ global lepton number symmetry.

In our model the full symmetry ${\mathcal {G}}$ experiences the following spontaneous symmetry breaking chain:

$$\begin{aligned}&{\mathcal {G}}=SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}\times U(1)_{L_{g}}\times S_{3}\times Z_{2}{\xrightarrow {v_{\chi },v_{\xi },v_{\varphi },} } \nonumber \\&\quad SU(3)_{C}\times SU(2)_{L}\times U(1)_{L_{g}}{ \xrightarrow {v_\eta ,v_{\rho }}} \nonumber \\&\quad SU(3)_{C}\times U(1)_{Q}\times U(1)_{L_{g}}{\xrightarrow {v_\sigma ,v_{\sigma _3}}} \nonumber \\&\quad SU(3)_{C}\times U(1)_{Q}, \end{aligned}$$

(3)

where the different symmetry breaking scales fulfill the hierarchy:

$$\begin{aligned} v_{\chi }\sim v_{\xi }\sim v_{\varphi }\gg v_{\eta },v_{\rho }\gg v_{\sigma }\sim v_{\sigma _{3}}, \end{aligned}$$

(4)

with $v_{\eta }^{2}+v_{\rho }^{2}=v^{2}$, $v=246$ GeV. We assume that the scale $v_{\chi }$ of spontaneous $\ SU(3)_{L}\times U(1)_{X}$ gauge symmetry breaking is about 10 TeV or larger in order to keep consistency with the collider constraints [104], the constraints from the experimental data on K, D and B-meson mixings [105] and $B_{s,d}\rightarrow \mu ^{+}\mu ^{-}$, $B_{d}\rightarrow K^{*}(K)\mu ^{+}\mu ^{-}$ decays [9, 106,107,108,109]

In principle, the hierarchical VEV pattern (4), being unprotected by any symmetry, can be affected by large radiative corrections. The common remedy against this issue is to assume that our model is embedded into a more fundamental setup with additional symmetries protecting the hierarchy up to the Planck scale. The well-known examples of such setups are supersymmetry and warped five-dimensions. Formulation of the corresponding ultraviolet completion is beyond the scope of the present paper and will be done elsewhere. One can also be concerned about the classical stability of the scalar potential at the vacuum configuration (4). The latter must belong to the minimum of the model scalar potential shown in Appendix B. This means that the scalar mass squared matrices in the vacuum ( 4) are positively definite. Having at our disposal a large number of free parameters in the scalar potential (B1) it is reasonable to expect that this condition can be easily satisfied in a wide range of the model parameter space. In Sect. 3 we show that this is true for the benchmark point (17) used for the analysis of $(g-2)_{e,\mu }$.

The $SU(3)_{L}$ triplet scalars $\chi$, $\eta$ and $\rho$ can be expanded around the minimum as follows:

$$\begin{aligned} \chi= & {} \begin{pmatrix} \frac{1}{\sqrt{2}}\left( \chi _{1R}^{0}+i\chi _{1I}^{0}\right) \\ \chi _{2}^{-} \\ \frac{1}{\sqrt{2}}(v_{\chi }+\xi _{\chi }\pm i\zeta _{\chi }) \end{pmatrix} ,\quad \nonumber \\ \eta= & {} \begin{pmatrix} \frac{1}{\sqrt{2}}(v_{\eta }+\xi _{\eta }\pm i\zeta _{\eta }) \\ \eta _{2}^{-} \\ \frac{1}{\sqrt{2}}\left( \eta _{3R}^{0}+i\eta _{3I}^{0}\right) \end{pmatrix} ,\quad \nonumber \\ \rho= & {} \begin{pmatrix} \rho _{1}^{+} \\ \frac{1}{\sqrt{2}}(v_{\rho }+\xi _{\rho }\pm i\zeta _{\rho }) \\ \rho _{3}^{+} \end{pmatrix} , \end{aligned}$$

(5)

Table 1 Scalar assignments under $SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}\times U(1)_{L_{g}}\times S_{3}\times Z_{2}$

Table 2 Fermion assignments under $SU(3)_{C}\times SU(3)_{L}\times U(1)_{X}\times U(1)_{L_{g}}\times S_{3}\times Z_{2}$. Here $B_{L,R}=\left( B_{1(L,R)},B_{2(L,R)}\right)$, $E_{L,R}=\left( E_{1(L,R)},E_{2(L,R)}\right)$, $L_{L}=\left( L_{1(L)},L_{2(L)}\right)$, $N_{R}=\left( N_{1(R)},N_{2(R)}\right)$, $n=1,2$ and $i=1,2,3$

The $SU(3)_L$ fermionic antitriplets and triplets are

$$\begin{aligned} Q_{nL}&= \begin{pmatrix} D_n \\ -U_n \\ J_n \\ \end{pmatrix} _L,\quad Q_{3L}= \begin{pmatrix} U_3 \\ D_3 \\ T \\ \end{pmatrix} _L,\quad L_{iL}= \begin{pmatrix} \nu _{i} \\ l_{i} \\ \nu _{i}^{c} \\ \end{pmatrix} _L,\quad \nonumber \\ n&=1,2,~i=1,2,3. \end{aligned}$$

(6)

where $l_{1,2,3}=e,\mu ,\tau$.

With the field assignment specified in Tables 1 and 2, the following quark and lepton Yukawa terms arise:

$$\begin{aligned} -{\mathcal {L}}_{Y}^{(q)}= & {} y_{T}{\overline{Q}}_{3L}\chi T_{R}+y_{J}\left( {\overline{Q}}_{L}\chi ^{*}J_{R}\right) _{{\mathbf {1}}}+y_{U}{\overline{Q}} _{3L}\eta U_{3R} \nonumber \\&+m_{\widetilde{T}}\left( \overline{\widetilde{T}}_{L} \widetilde{T}_{R}\right) _{{\mathbf {1}}}+m_{B}\left( {\overline{B}} _{L}B_{R}\right) _{{\mathbf {1}}} +m_{B_{3}}{\overline{B}}_{3L}B_{3R} \nonumber \\&+x_{T}\left( {\overline{Q}}_{L}\rho ^{*}\widetilde{T}_{R}\right) _{ {\mathbf {1}}}+\sum _{n=1}^{2}z_{n}^{(U)}\left( \overline{\widetilde{T}}_{L}\xi \right) _{{\mathbf {1}}^{\prime }}U_{nR} \nonumber \\&+x_{B}\left( {\overline{Q}}_{L}\eta ^{*}B_{R}\right) _{{\mathbf {1}}}+\sum _{j=1}^{3}z_{j}^{\left( D\right) }\left( {\overline{B}}_{L}\xi \right) _{{\mathbf {1}}^{\prime }}D_{jR} \nonumber \\&+y_{B}{\overline{Q}}_{3L}\rho B_{3R}+\sum _{j=1}^{3}x_{j}^{\left( D\right) } {\overline{B}}_{3L}\varphi D_{jR}+H.c, \end{aligned}$$

(7)

$$\begin{aligned} -{\mathcal {L}}_{Y}^{(l)}= & {} x_{E}\left( {\overline{L}}_{L}\rho E_{R}\right) _{ {\mathbf {1}}}+\sum _{j=1}^{3}z_{j}^{(l)}\left( {\overline{E}}_{L}\xi \right) _{ {\mathbf {1}}^{\prime }}l_{jR}\nonumber \\&+y_{E}{\overline{L}}_{3L}\rho E_{3R}+\sum _{j=1}^{3}x_{j}^{(l)}{\overline{E}}_{3L}\varphi l_{jR} \nonumber \\&+m_{E}\left( {\overline{E}}_{L}E_{R}\right) _{{\mathbf {1}}}+m_{E_{3}}{\overline{E}}_{3L}E_{3R} \nonumber \\&+x_{N}\left( {\overline{L}}_{L}\chi N_{R}\right) _{\mathbf {{\mathbf {1}}}}+y_{N} {\overline{L}}_{3L}\chi N_{3R} \nonumber \\&+h_{1N}\left( N_{R}\overline{N_{R}^{C}}\right) _{ \mathbf {{\mathbf {2}}}}\sigma +h_{2N}\left( N_{R}\overline{N_{3R}^{C}}\sigma \right) _{{\mathbf {1}}^{\prime }} \nonumber \\&+h_{3N}\left( N_{R}\overline{N_{R}^{C}}\right) _{\mathbf {{\mathbf {1}}} }\sigma _{3}+h_{4N}N_{3R}\overline{N_{3R}^{C}}\sigma _{3}+H.c.\nonumber \\ \end{aligned}$$

(8)

We consider the following VEV configurations for the $S_{3}$ doublets:

$$\begin{aligned} \left\langle \xi \right\rangle =v_{\xi }\left( 1,0\right) ,\quad \left\langle \sigma \right\rangle =\left( v_{\sigma _{1}},v_{\sigma _{2}}\right) , \end{aligned}$$

(9)

which are consistent with the scalar potential minimization equations for a large region of parameter space [20, 90, 110].

Fig. 1

Leading Loop Feynman diagrams contributing to the muon and electron anomalous magnetic moments. Here $E_{1,2}$, are components of the $S_{3}$ -doublet and $j=1,2,3,4$

3 Muon and electron anomalous magnetic moments

The current experimental data on the anomalous dipole magnetic moments of electron and muon $a_{e,\mu }=(g_{e,\mu }-2)/2$ show significant deviation from their SM values

$$\begin{aligned} \Delta a_{\mu }= & {} a_{\mu }^{\mathrm {exp}}-a_{\mu }^{\mathrm {SM}} \nonumber \\= & {} \left( 26.1\pm 8\right) \times 10^{-10}\qquad \quad \,[111--117] \end{aligned}$$

(10)

$$\begin{aligned} \Delta a_{e}= & {} a_{e}^{\mathrm {exp}}-a_{e}^{\mathrm {SM}} \nonumber \\= & {} (-0.88\pm 0.36)\times 10^{-12}\qquad 118 \end{aligned}$$

(11)

Here we analyze predictions of our model for these observables. The leading contributions to $\Delta a_{e,\mu }$ arising in the model are shown in Fig.  1. The diagrams involve the electrically neutral physical CP even $H^{0}_{i}$ ($i=1,2,3,4$) and CP odd $A^{0}$ scalar as well as heavy exotic charged $E_{L,R}$ leptons. The physical CP even scalars arise from the combinations of $\xi _{\rho }$, $\xi _{1R}$, $\xi _{2R}$, $\varphi _{R}$ whereas the CP odd scalar corresponds to $\zeta _{\rho }$. By $\xi _{1R,2R}$ we denote real part of the two components of the scalar $S_{3}$-doublet gauge singlet $\xi$. Similary, the real part of the scalar $S_{3}$-singlet gauge singlet $\varphi$ is denoted by $\varphi _{R}$. Analogously, $E_{1,2}$ are two components of the leptonic $S_{3}$-doublet gauge singlet $E_{L,R}$. The fields $\xi _{\rho }$ and $\zeta _{\rho }$ are contained in the $SU(3)_{L}$ scalar triplet $\rho$, which interacts with l , the second component of the leptonic triplet $L_{L}$. It is worth mentioning that, in view of the large amount of parametric freedom of the model scalar potential in Eq. (B1), we are restricting to a particular benchmark scenario were the $SU\left( 3\right) _{L}$ scalar triplet $\rho$ and the gauge singlet scalars $\xi$ and $\varphi$ do not feature mixings with the remaining scalar fields $\eta$, $\sigma$ and $\sigma _{3}$. Such benchmark scenario is consistent with the decoupling limit where the CP even neutral component of the $SU\left( 3\right) _{L}$ scalar triplet $\eta$ mostly corresponds to the 126 GeV SM like Higgs boson. Another motivation for such benchmark scenario is the fact that the VEV of the $SU\left( 3\right) _{L}$ scalar triplet $\chi$ is much larger than the VEV of the $SU\left( 3\right) _{L}$ scalar triplet $\rho$, thus allowing to neglect the mixing angles between those fields since they are suppressed by the ratios of their VEVs, as follows from the method of recursive expansion of Ref. [119]. Due to the same argument, the mixing angles of the $\rho$, $\xi$ and $\varphi$ scalar fields with the gauge singlet scalars $\sigma$ and $\sigma _{3}$ can be neglected as well.

In this framework, the scalar potential terms contributing to the Yukawa couplings of the fermions $E_1$ and $E_2$ with the scalar fields are shown in Appendix C. Let us note the the following peculiar pattern of mixing in the scalar sector. The fields $\rho$, $\xi$ and $\varphi$ do not mix with $\eta$, $\sigma$ and $\sigma _{3}$, while $\varphi$ mix with $\zeta _{\rho }$ through the the complex parameter $\kappa$ in the scalar potential (B1). In view of this we find that the scalar mass matrix in the basis $\xi _{\rho }$, $\xi _{1R}$, $\xi _{2R}$, $\varphi$, and $\zeta _{\rho }$ has the form:

$$\begin{aligned} {\mathbf {M}}^{2}&= \begin{pmatrix} m^{2}_{11} &{} m^{2}_{12} &{} 0 &{} m^{2}_{14} &{} 0 \\ m^{2}_{21} &{} m^{2}_{22} &{} m^{2}_{23} &{} m^{2}_{24} &{} 0 \\ 0 &{} m_{32} &{} m_{33} &{} m_{34} &{} 0 \\ m^{2}_{41} &{} m^{2}_{42} &{} m^{2}_{43} &{} m^{2}_{44} &{} m^{2}_{45} \\ 0 &{} 0 &{} 0 &{} m^{2}_{54} &{} m^{2}_{55} \\ &{} &{} &{} &{} \end{pmatrix} \end{aligned}$$

(12)

with the matrix elements $m^{2}_{ij}$ given in Appendix C. Once the basis is changed by a rotation matrix R, the physical scalar field masses $m_{H^{0}_{1}}$, $m_{H^{0}_{2}}$, $m_{H^{0}_{3}}$, $m_{H^{0}_{5}}$ and $m_{H^{0}_{A}}$ can be found numerically.

Thus, in our model the muon and electron anomalous magnetic moments are given by:

$$\begin{aligned} \Delta a_{\mu ,e}=\sum \limits _{i=1}^{4}\sum \limits _{\Phi =H^{0}_{i},A^{0}}\Delta a_{\mu ,e}(\Phi ), \end{aligned}$$

(13)

The analytical form for the neutral scalar contribution at one loop to $\Delta a_{\mu ,e}$ can be found in [22, 120,121,122]. Using these results we write the contributions of the neutral scalars $\Phi =H^{0},A^{0}$ as follows:

$$\begin{aligned} \Delta a_{\mu }= & {} w_{\mu }^{2}\frac{m_{\mu }^{2}}{8\pi ^{2}}\left\{ \sum \limits _{i=1}^{4}\left( R^{T}\right) _{1i}\left( R^{T}\right) _{2i} \frac{G_{S}^{\left( l\right) }\left( m_{E_{2}},m_{H_{i}^{0}}\right) }{ m_{H_{i}^{0}}^{2}} \right. \nonumber \\&\left. +\left( R^{T}\right) _{55}\left( R^{T}\right) _{25}\frac{ G_{P}^{\left( l\right) }\left( m_{E_{2}},m_{A^{0}}\right) }{m_{A^{0}}^{2}} \right\} \end{aligned}$$

(14)

$$\begin{aligned} \Delta a_{e}= & {} w_{e}^{2}\frac{m_{\mu }^{2}}{8\pi ^{2}}\left\{ \sum \limits _{i=1}^{4}\left( R^{T}\right) _{1i}\left( R^{T}\right) _{3i} \frac{G_{S}^{\left( l\right) }\left( m_{E_{1}},m_{H_{i}^{0}}\right) }{ m_{H_{i}^{0}}^{2}} \right. \nonumber \\&\left. +\left( R^{T}\right) _{55}\left( R^{T}\right) _{35}\frac{ G_{P}^{\left( l\right) }\left( m_{E_{1}},m_{A^{0}}\right) }{m_{A^{0}}^{2}} \right\} \end{aligned}$$

(15)

where the loop function is given by:

$$\begin{aligned}&G_{S,P}^{(l)}(m_{E}, m_{\Phi } ) \nonumber \\&\quad =\int _{0}^{1}dx\frac{x^{2}(1-x\pm \epsilon _{lE})}{(1-x)(1-x\lambda _{l\Phi }^{2})+x\epsilon _{lE}^{2}\lambda _{l\Phi }^{2}},\qquad \Phi =H^{0},A^{0} \end{aligned}$$

(16)

with $l=e,\mu$ and $\lambda _{l\Phi }=m_{l}/m_{\Phi }$, $\epsilon _{eE}=m_{E_{1}}/m_{e}$, $\epsilon _{\mu E}=m_{E_{2}}/m_{\mu }$. Besides that, the plus and minus signs for the loop function $G_{S,P}(\Phi )$ of Eq. (16) stands for the scalar (CP-even) and pseudoscalar (CP-odd) contributions, respectively. The quantities $w_{l}$ ($l=e,\mu$) are the Yukawa couplings for the interaction $w_{l}{\overline{E}}l\,\Phi$.

The experimental values of the muon and electron anomalous magnetic moments shown in Eqs. (10) and (11) can be successfully reproduced at $2\sigma$ level for the following benchmark point:

$$\begin{aligned} v_{\eta }&= v_{\rho } \approx 174\,\text {GeV}&v_{\chi }&\approx 2851\,\text { GeV}&v_{\xi }&\approx 1414\,\text {GeV} \nonumber \\ v_{\phi }&\approx 6992\,\text {GeV}&\kappa _r&\approx -0.630&\kappa _i&\approx -0.614 \nonumber \\ \lambda _{2}&= \lambda _{10} \approx 7.251&\lambda _{12}&= \lambda _{13} \approx 0.310&\nonumber \\ \lambda _{18}&= \lambda _{34} \approx -0.264&\lambda _{35}&= \lambda _{38} \approx -0.229 \end{aligned}$$

(17)

The scalar and charged exotic leptons masses along with the Yukawa couplings are

$$\begin{aligned} m_{H^0_1}&\approx 5786\,\text {GeV}&m_{H^0_2}&\approx 5338\,\text {GeV}&m_{H^0_3}&\approx 2750\,\text {GeV} \nonumber \\ m_{H^0_4}&\approx 2498\,\text {GeV}&m_{A^0}&\approx 1100 \,\text {GeV}&m_{E_1}&\approx 611\,\text {GeV} \nonumber \\ m_{E_2}&\approx 1368\,\text {GeV}&w_{\mu }&\approx 0.228&w_{e}&\approx 1.719 \end{aligned}$$

(18)

Note that this benchmark point locates in the domain of the model parameter space corresponding to the minimum of the scalar potential due to the fact that all the scalar masses are real (see also Appendix C). In this benchmark point the muon and electron $(g-2)$ -experimental anomalies have the values

$$\begin{aligned} \Delta a_{\mu }&= 2.68714 \times 10^{-9} \end{aligned}$$

(19)

$$\begin{aligned} \Delta a_{e}&= -8.64531 \times 10^{-13} \end{aligned}$$

(20)

The opposite signs of these quantities is due to the pseudo scalar $A^{0}$ contributions to the loops in Fig. 1 leading to the minus sign in the term $-\epsilon _{lE}$ of the loop function (16). Note that $E_{2}$ and $E_{1}$ contribute separately to the muon and electron $(g-2)$, respectively, without any cross-contributions. Thus, selecting appropriate values for the exotic lepton masses $m_{E_{1,2}}$ we can accommodate the experimental sign difference (19), (20). The fact that $m_{e}\ll m_{\mu }$ makes the required sign difference valid in a wide range of the model parameter space. To show this, let us vary the model parameters within 15% around the benchmark point (17) and the charged exotic lepton masses in a range from 200 to 1200 GeV. The resulting $\Delta a_{\mu ,e}-m_{E_{2,1}}$ scatter plots are shown in Fig. 2. As can be seen, the model indeed can explain the experimental values of muon and electron anomalous magnetic moments simultaneously in a wide range of its parameter space.

Fig. 2

Correlation plots of the $\Delta a_{\mu , e}$ and the mass of the exotic fermion $m_{E_{2,1}}$ respectively at $1\sigma$ (red), $2\sigma$ (brown) and $3\sigma$ (purple)

4 Quark masses and mixings

From the quark Yukawa interactions in Eq. (7), we find that the up-type mass matrix in the basis $({\overline{u}}_{1L},{\overline{u}}_{2L}, {\overline{u}}_{3L},{\overline{T}}_{L},\overline{\widetilde{T}}_{1L},\overline{ \widetilde{T}}_{2L})$ versus $(u_{1R},u_{2R},u_{3R},T_{R},\widetilde{T}_{1R}, \widetilde{T}_{2R})$ is given by:

$$\begin{aligned} M_{U}= & {} \left( \begin{array}{cccc} \Delta _{U} &{} 0_{2\times 1} &{} 0_{2\times 1} &{} A_{U} \\ 0_{1\times 2} &{} m_{t} &{} 0 &{} 0_{1\times 2} \\ 0_{1\times 2} &{} 0 &{} m_{T} &{} 0_{1\times 2} \\ B_{U} &{} 0_{2\times 1} &{} 0_{2\times 1} &{} \widetilde{M}_{T} \end{array} \right) ,\quad \nonumber \\ A_{U}= & {} x_{T}\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \end{array} \right) \frac{v_{\rho }}{\sqrt{2}},\quad B_{U}=\left( \begin{array}{cc} 0 &{} 0 \\ z_{1}^{(U)} &{} z_{2}^{(U)} \end{array} \right) v_{\xi }, \nonumber \\ m_{t}= & {} y_{U}\frac{v_{\eta }}{\sqrt{2}}=a_{3}^{(U)}\frac{v}{\sqrt{2}}, \quad \widetilde{M}_{T}=m_{\widetilde{T}}\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \end{array} \right) ,\quad \nonumber \\ \Delta _{U}= & {} \left( \begin{array}{cc} \varepsilon _{11}^{(U)} &{} \varepsilon _{12}^{(U)} \\ \varepsilon _{21}^{(U)} &{} \varepsilon _{22}^{(U)} \end{array} \right) \frac{v_{\rho }}{\sqrt{2}}, \nonumber \\ \varepsilon _{1n}^{(U)}= & {} \frac{1}{16\pi ^{2}}\frac{\lambda _{\rho ^{\dagger }\rho \xi ^{2}}\lambda _{\xi ^{3}\varphi }x_{T}z_{n}^{(U)}v_{\varphi }v_{\xi }^{2}}{m_{T}m_{\xi _{2}}^{2}}\left[ C_{0}\left( \frac{m_{\xi _{\eta }}}{m_{B}},\frac{m_{\text {Re}\xi _{2}}}{m_{B} }\right) \right. \nonumber \\&\left. -C_{0}\left( \frac{m_{\zeta _{\eta }}}{m_{B}},\frac{m_{\text {Im}\xi _{2}}}{m_{B}}\right) \right] ,\quad n=1,2, \nonumber \\ \varepsilon _{2n}^{(U)}= & {} \frac{1}{16\pi ^{2}}\frac{\lambda _{\rho ^{\dagger }\rho \xi ^{2}}x_{T}z_{n}^{(U)}v_{\xi }}{m_{T}}\left[ C_{0}\left( \frac{m_{\xi _{\eta }}}{m_{B}},\frac{m_{\text {Re}\xi _{2}}}{m_{B}}\right) \right. \nonumber \\&\left. -C_{0}\left( \frac{m_{\zeta _{\eta }}}{m_{B}},\frac{m_{\text {Im}\xi _{2}}}{ m_{B}}\right) \right] , \end{aligned}$$

(21)

whereas the down type quark mass matrix written in the basis $({\overline{d}}_{1L},{\overline{d}}_{2L},{\overline{d}}_{3L},{\overline{J}}_{1L}, {\overline{J}}_{2L},{\overline{B}}_{1L},{\overline{B}}_{2L},{\overline{B}}_{3L})$-$(d_{1R},d_{2R},d_{3R},J_{1R},J_{2R},B_{1R},B_{2R},B_{3R})$ takes the form:

$$\begin{aligned} M_{D}= & {} \left( \begin{array}{ccc} \Delta _{D} &{} 0_{3\times 2} &{} A_{D} \\ 0_{2\times 3} &{} M_{J} &{} 0_{2\times 3} \\ B_{D} &{} 0_{3\times 2} &{} M_{B} \end{array} \right) ,~ A_{D}=\left( \begin{array}{ccc} x_{B}\frac{v_{\eta }}{\sqrt{2}} &{} 0 &{} 0 \\ 0 &{} x_{B}\frac{v_{\eta }}{\sqrt{2}} &{} 0 \\ 0 &{} 0 &{} y_{B}\frac{v_{\rho }}{\sqrt{2}} \end{array} \right) ,~\nonumber \\ B_{D}= & {} \left( \begin{array}{ccc} 0 &{} 0 &{} 0 \\ z_{1}^{(D)}v_{\xi } &{} z_{2}^{(D)}v_{\xi } &{} z_{3}^{(D)}v_{\xi } \\ x_{1}^{(D)}v_{\varphi } &{} x_{2}^{(D)}v_{\varphi } &{} x_{3}^{(D)}v_{\varphi } \end{array} \right) , \nonumber \\ M_{J}= & {} y^{\left( J\right) }\frac{v_{\chi }}{\sqrt{2}}\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \end{array} \right) ,\qquad M_{B}=\left( \begin{array}{ccc} m_{B} &{} 0 &{} 0 \\ 0 &{} m_{B} &{} 0 \\ 0 &{} 0 &{} m_{B_{3}} \end{array} \right) ,\qquad \nonumber \\ \Delta _{D}= & {} \left( \begin{array}{ccc} \varepsilon _{11}^{(D)} &{} \varepsilon _{12}^{(D)} &{} \varepsilon _{13}^{(D)} \\ \varepsilon _{21}^{(D)} &{} \varepsilon _{22}^{(D)} &{} \varepsilon _{23}^{(D)} \\ \varepsilon _{31}^{(D)} &{} \varepsilon _{32}^{(D)} &{} \varepsilon _{33}^{(D)} \end{array} \right) \frac{v_{\rho }}{\sqrt{2}}, \nonumber \\ \varepsilon _{1i}^{(D)}= & {} \frac{1}{16\pi ^{2}}\frac{\lambda _{\rho ^{\dagger }\rho \xi ^{2}}\lambda _{\xi ^{3}\varphi }x_{B}z_{i}^{(D)}v_{\varphi }v_{\xi }^{2}v_{\eta }}{m_{B}m_{\xi _{2}}^{2}v_{\rho }}C_{0}\left( \frac{m_{\xi _{\eta }}}{m_{B}},\frac{m_{\text { Re}\xi _{2}}}{m_{B}}\right) , \nonumber \\ \varepsilon _{2i}^{(D)}= & {} \frac{1}{16\pi ^{2}}\frac{\lambda _{\rho ^{\dagger }\rho \xi ^{2}}x_{B}z_{i}^{(D)}v_{\xi }v_{\eta }}{m_{B}v_{\rho }} C_{0}\left( \frac{m_{\xi _{\eta }}}{m_{B}},\frac{m_{\text {Re}\xi _{2}}}{m_{B} }\right) , \nonumber \\ \varepsilon _{3i}^{(D)}= & {} \frac{1}{16\pi ^{2}}\frac{\lambda _{\rho ^{\dagger }\rho \varphi ^{2}}x_{B}z_{i}^{(D)}v_{\varphi }}{m_{B}}C_{0}\left( \frac{m_{\xi _{\rho }}}{m_{B}},\frac{m_{\text {Re}\varphi }}{m_{B}}\right) , \quad \nonumber \\ i= & {} 1,2,3, \end{aligned}$$

(22)

where as seen from Eqs. (21) and (22), the $\Delta _U$ and $\Delta _D$ submatrices are generated at one loop level. The one loop level Feynman diagrams generating the $\Delta _U$ and $\Delta _D$ submatrices are shown in Fig. 3. In addition, the following function has been introduced:

$$\begin{aligned}&C_{0}\left( {\widehat{m}}_1 ,{\widehat{m}}_2 \right) \nonumber \\&\quad =\frac{1}{\left( 1-\widehat{ m}_1 ^2 \right) \left( 1-{\widehat{m}}_2 ^2 \right) \left( {\widehat{m}}_1 ^2 - {\widehat{m}}_2 ^2 \right) }\left\{ {\widehat{m}}_1 ^2 {\widehat{m}}_2 ^2 \ln \left( \frac{{\widehat{m}}_1 ^2 }{{\widehat{m}}_2 ^2 }\right) \right. \nonumber \\&\left. \qquad -{\widehat{m}}_1 ^2 \ln {\widehat{m}}_1 ^2 +{\widehat{m}}_2 ^2 \ln {\widehat{m}}_2 ^2 \right\} . \end{aligned}$$

(23)

Fig. 3

One-loop Feynman diagrams contributing to the entries of the SM quark mass matrices. Here, $n=1,2$ and $i=1,2,3$

As seen from Eqs. (21) and (22), the exotic heavy vector like quarks mix with the SM quarks lighter than top quark. The masses of these exotic quarks are assumed to be much larger than the $SU(3) _L\times U(1) _X$ symmetry breaking scale. As a result, charm, bottom and strange quarks acquire their masses from the tree-level Universal seesaw mechanism, while the masses of the up and down quarks are generated by the one-loop radiative seesaw mechanism. Thus, for the SM quarks we obtain the following mass matrices:

$$\begin{aligned} \widetilde{M}_U= & {} \left( \begin{array}{cc} \Delta _U +A_U M_{\widetilde{T}}^{-1}B_U &{} 0_{2\times 1} \\ 0_{1\times 2} &{} m_{t} \end{array} \right) \nonumber \\= & {} \left( \begin{array}{ccc} \varepsilon _{11}^{(U) }\frac{v_\rho }{\sqrt{2}} &{} \varepsilon _{12}^{(U) } \frac{v_\rho }{\sqrt{2}} &{} 0 \\ \varepsilon _{21}^{(U) }\frac{v_\rho }{\sqrt{2}}+\frac{x_{T}z_1 ^{(U) }v_\xi v_\rho }{\sqrt{2}m_{\widetilde{T}}} &{} \varepsilon _{22}^{(U) }\frac{v_\rho }{ \sqrt{2}}+\frac{x_{T}z_2 ^{(U) }v_\xi v_\rho }{\sqrt{2}m_{\widetilde{T}}} &{} 0 \\ 0 &{} 0 &{} m_{t} \end{array} \right) , \end{aligned}$$

(24)

$$\begin{aligned} \widetilde{M}_D= & {} \Delta _D +A_D M_B ^{-1}B_D \nonumber \\= & {} \left( \begin{array}{ccc} \varepsilon _{11}^{(D) }\frac{v_\rho }{\sqrt{2}} &{} \varepsilon _{12}^{(D) } \frac{v_\rho }{\sqrt{2}} &{} \varepsilon _{13}^{(D) }\frac{v_\rho }{\sqrt{2}} \\ \varepsilon _{21}^{(D) }\frac{v_\rho }{\sqrt{2}}+x_B z_1 ^{(D) }\frac{v_\xi v_\eta }{\sqrt{2}m_B } &{} \varepsilon _{22}^{(D) }\frac{v_\rho }{\sqrt{2}}+x_B z_2 ^{(D) }\frac{v_\xi v_\eta }{\sqrt{2}m_B } &{} \varepsilon _{23}^{(D) } \frac{v_\rho }{\sqrt{2}}+x_B z_3 ^{(D) }\frac{v_\xi v_\eta }{\sqrt{2}m_B } \\ \varepsilon _{31}^{(D) }\frac{v_\rho }{\sqrt{2}}+y_B x_1 ^{(D) }\frac{ v_{\varphi }v_\rho }{\sqrt{2}m_{B_3 }} &{} \varepsilon _{32}^{(D) }\frac{v_\rho }{\sqrt{2}}+y_B x_2 ^{(D) }\frac{v_{\varphi }v_\rho }{\sqrt{2}m_{B_3 }} &{} \varepsilon _{33}^{(D) }\frac{v_\rho }{\sqrt{2}}+y_B x_3 ^{(D) }\frac{ v_{\varphi }v_\rho }{\sqrt{2}m_{B_3 }} \end{array} \right) . \end{aligned}$$

(25)

These mass matrices contain several model parameters. While free, they still satisfy certain conditions in our model. In fact, vev’s $v_{\xi ,\eta ,\rho }$ obey the inequality (4) expressing the hierarchy of symmetry breaking in our model. The $\varepsilon _{ij}^{U,D}$ parameters, defined in Eqs. (21) and (22), contain typical loop suppression and specific dependence on the vev’s, exotic masses, the Yukawas and a quartic coupling. We require the latter to satisfy the perturbativity condition. With this in mind we can speak about natural values of the matrix elements corresponding to the values of the model parameters in a range not involving an ad hoc hierarchy of the dimensionless couplings. Let us show that within this natural range the model accommodates the observable values of the SM quark masses and mixings. To this end we consider a particular natural benchmark scenario consistent with the above-mentioned conditions. We choose:

$$\begin{aligned} v_\xi= & {} \lambda ^{4}\frac{vm_{\widetilde{T}}}{v_\rho }=\lambda ^5\frac{vm_B }{v_\eta },\quad v_{\varphi }=\lambda ^3 \frac{vm_{B_3 }}{v_\rho }, \quad \nonumber \\ \varepsilon _{nm}^{(U) }= & {} b_{nm}^{(U) }\lambda ^{8}\frac{v}{ v_\rho }, \nonumber \\ \varepsilon _{ij}^{(D) }= & {} b_{ij}^{(D) }\lambda ^{7}\frac{v}{v_\rho }, \quad i,j=1,2,3;\, n,m=1,2, \end{aligned}$$

(26)

where $v=\sqrt{v_\rho ^2 +v_\eta ^2 }=246$ GeV is the electroweak symmetry breaking scale. We use the Wolfenstein parameter $\lambda =0.225$ for characterization of the hierarchy between the parameters defining mass matrix elements. As discussed above, we consider the hierarchy, which stems from the model structure rather than from strong tuning of the dimensionless couplings. Then the coefficients $b_{nm}^{(U) }$ and $b_{ij}^{(D) }$, constructed from the Yukawa and quartic couplings, are ${\mathcal {O}}(1)$ -numbers. In the scenario (26) the exotic quarks ${\tilde{T}}$ and B are heavier than the scale of the first stage of the symmetry breaking (3). As we mentioned earlier, these exotic quarks must be very heavy for the Universal Seesaw mechanism to operate in our model.

Thus, in the benchmark scenario (26) the SM quark mass matrices take the form:

$$\begin{aligned} \widetilde{M}_U= & {} \left( \begin{array}{ccc} b_{11}^{(U) }\lambda ^{8} &{} b_{12}^{(U) }\lambda ^{8} &{} 0 \\ b_{21}^{(U) }\lambda ^{7}+a_{21}^{(U) }\lambda ^{4} &{} b_{22}^{(U) }\lambda ^{7}+a_{22}^{(U) }\lambda ^{4} &{} 0 \\ 0 &{} 0 &{} m_{t} \end{array} \right) \frac{v}{\sqrt{2}} \nonumber \\= & {} \left( \begin{array}{ccc} b_{11}^{(U) }\lambda ^{8} &{} b_{12}^{(U) }\lambda ^{8} &{} 0 \\ c_{21}^{(U) }\lambda ^{4} &{} c_{22}^{(U) }\lambda ^{4} &{} 0 \\ 0 &{} 0 &{} m_{t} \end{array} \right) \frac{v}{\sqrt{2}}, \end{aligned}$$

(27)

$$\begin{aligned} \widetilde{M}_D= & {} \left( \begin{array}{ccc} b_{11}^{(D) }\lambda ^{7} &{} b_{12}^{(D) }\lambda ^{7} &{} b_{13}^{(D) }\lambda ^{7} \\ b_{21}^{(D) }\lambda ^{7}+a_{21}^{(D) }\lambda ^5 &{} b_{22}^{(D) }\lambda ^{7}+a_{22}^{(D) }\lambda ^5 &{} b_{23}^{(D) }\lambda ^{7}+a_{23}^{(D) }\lambda ^5 \\ b_{31}^{(D) }\lambda ^{7}+a_{31}^{(D) }\lambda ^3 &{} b_{32}^{(D) }\lambda ^{7}+a_{32}^{(D) }\lambda ^3 &{} b_{33}^{(D) }\lambda ^{7}+a_{33}^{(D) }\lambda ^3 \end{array} \right) \frac{v}{\sqrt{2}} \nonumber \\= & {} \left( \begin{array}{ccc} b_{11}^{(D) }\lambda ^7 &{} b_{12}^{(D) }\lambda ^{7} &{} b_{13}^{(D) }\lambda ^{7} \\ c_{21}^{(D) }\lambda ^5 &{} c_{22}^{(D) }\lambda ^5 &{} c_{23}^{(D) }\lambda ^5 \\ c_{31}^{(D) }\lambda ^3 &{} c_{32}^{(D) }\lambda ^3 &{} c_{33}^{(D) }\lambda ^3 \end{array} \right) \frac{v}{\sqrt{2}}, \end{aligned}$$

(28)

where

$$\begin{aligned} a_{21}^{(U) }&=x_{T}z_1 ^{(U) },\qquad a_{22}^{(U) }=x_{T}z_2 ^{(U) }, \nonumber \\ a_{2i}^{(D) }&=x_B z_{i}^{(D) },\qquad a_{3i}^{(D) }=y_B x_{i}^{(D) }, \qquad i=1,2,3. \end{aligned}$$

(29)

The model has 13 dimensionless parameters in the quark sector. This allows us to reproduce precisely the central experimental values of 10 quark observables, shown in Table 3. The corresponding values of the model parameters are:

$$\begin{aligned} b_{11}^{(U) }= & {} c_{21}^{(U) }=1,\qquad \quad b_{12}^{(U) }\simeq 2.773, \qquad \quad \nonumber \\ b_{22}^{(U) }\simeq & {} 1.001,\qquad a_3 ^{(U) }\simeq 0.989, \nonumber \\ b_{11}^{(D) }\simeq & {} -1.335+0.929i,~b_{12}^{(D) }\simeq 1.217+1.314 i,~\nonumber \\ b_{13}^{(D) }\simeq & {} 2.112-0.929i, \nonumber \\ c_{21}^{(D) }\simeq & {} -0.869,\qquad \quad c_{22}^{(D) }\simeq -0.438, \qquad \nonumber \\ c_{13}^{(D) }\simeq & {} 0.860, \nonumber \\ c_{31}^{(D) }\simeq & {} -0.707,\qquad \quad c_{32}^{(D) }\simeq -1.001, \qquad \nonumber \\ c_{33}^{(D) }\simeq & {} 0.707. \end{aligned}$$

(30)

An important point for us is that all these values are of the order of one. As we previously discussed, this means that the hierarchy of the quark masses and mixings originate in our model from its internal structure – symmetries and field content – without the need for strong tuning the dimensionless couplings.

Table 3 Experimental $M_{Z}$-scale values of the quark masses [123, 124] and CKM parameters [125]

5 Charged lepton masses and mixings

From the charged lepton Yukawa interactions in Eq. (8) we find the charged lepton mass matrix $M_{l}$ in the basis $( {\overline{l}}_{1L}, {\overline{l}}_{2L},{\overline{l}}_{3L},{\overline{E}}_{1L}, {\overline{E}}_{2L}, {\overline{E}}_{3L})$ versus $(l_{1R},l_{2R},l_{3R},E_{1R},E_{2R},E_{3R})$ given by:

$$\begin{aligned} M_l= & {} \left( \begin{array}{cc} \Delta _l &{} A_l \\ B_l &{} \widetilde{M}_E \end{array} \right) ,\quad \Delta _l=\left( \begin{array}{ccc} \varepsilon _{11}^{(l) } &{} \varepsilon _{12}^{(l) } &{} \varepsilon _{13}^{(l) } \\ \varepsilon _{21}^{(l) } &{} \varepsilon _{22}^{(l) } &{} \varepsilon _{23}^{(l) } \\ \varepsilon _{31}^{(l) } &{} \varepsilon _{32}^{(l) } &{} \varepsilon _{33}^{(l) } \end{array} \right) \frac{v_\rho }{\sqrt{2}},\quad \nonumber \\ A_l= & {} \left( \begin{array}{ccc} x_E &{} 0 &{} 0 \\ 0 &{} x_E &{} 0 \\ 0 &{} 0 &{} y_E \end{array} \right) \frac{v_\rho }{\sqrt{2}}, \nonumber \\ B_l= & {} \left( \begin{array}{ccc} 0 &{} 0 &{} 0 \\ -z_1 ^{(l) }v_\xi &{} -z_2 ^{(l) }v_\xi &{} -z_3 ^{(l) }v_\xi \\ x_1 ^{(l) }v_{\varphi } &{} x_2 ^{(l) }v_{\varphi } &{} x_3 ^{(l) }v_{\varphi } \end{array} \right) ,\quad \nonumber \\ \widetilde{M}_E= & {} \left( \begin{array}{ccc} m_E &{} 0 &{} 0 \\ 0 &{} m_E &{} 0 \\ 0 &{} 0 &{} m_{E_3 } \end{array} \right) , \nonumber \\ \varepsilon _{1i}^{(l) }= & {} \frac{1}{16\pi ^2 }\frac{\lambda _{\rho ^\dagger \rho \xi ^2 }\lambda _{\xi ^3 \varphi }x_E z_{i}^{(l) }v_{\varphi }v_\xi ^2 }{m_E m_{\xi _2 }^2 }\left[ C_{0}\left( \frac{m_{\xi _\rho }}{m_E },\frac{m_{ \text {Re}\xi _2 }}{m_E }\right) \right. \nonumber \\&\left. -C_{0}\left( \frac{m_{\zeta _\rho }}{m_E }, \frac{m_{\text {Im}\xi _2 }}{m_E }\right) \right] , \nonumber \\ \varepsilon _{2i}^{(l) }= & {} \frac{1}{16\pi ^2 }\frac{\lambda _{\rho ^\dagger \rho \xi ^2 }x_E z_{i}^{(l) }v_\xi }{m_E }\left[ C_{0}\left( \frac{m_{\xi _\rho }}{m_E },\frac{m_{\text {Re}\xi _2 }}{m_E }\right) \right. \nonumber \\&\left. -C_{0}\left( \frac{ m_{\zeta _\rho }}{m_E },\frac{m_{\text {Im}\xi _2 }}{m_E }\right) \right] , \nonumber \\ \varepsilon _{3i}^{(l) }= & {} \frac{1}{16\pi ^2 }\frac{\lambda _{\rho ^\dagger \rho \varphi ^2 }x_E z_{i}^{\left( E\right) }v_{\varphi }}{m_E }\left[ C_{0}\left( \frac{m_{\xi _\rho }}{m_E },\frac{m_{\text {Re}\varphi }}{m_E } \right) \right. \nonumber \\&\left. -C_{0}\left( \frac{m_{\zeta _\rho }}{m_E },\frac{m_{\text {Im}\varphi } }{m_E }\right) \right] ,\quad i=1,2,3. \end{aligned}$$

(31)

where as seen from Eq. (31), the $\Delta _l$ submatrix is generated at one loop level according to the Feynman diagrams shown in Fig. 4.

Fig. 4

One-loop Feynman diagrams contributing to the entries of the SM charged lepton mass matrix. Here $i=1,2,3$

As follows from Eq. (31), the very heavy vector like charged leptons mix with the SM charged leptons. The former are assumed to have masses much larger than the $SU(3) _L\times U(1) _X$ symmetry breaking scale. Therefore, analogously to the quark sector, the tau and muon masses are generated by the tree level Universal seesaw mechanism, while the electron mass arises from the one loop level radiative seesaw mechanism. Consequently, SM charged lepton mass matrix takes the form

$$\begin{aligned} \widetilde{M}_l= & {} \Delta _l+A_l\widetilde{M}_E ^{-1}B_l \nonumber \\= & {} \left( \begin{array}{ccc} \varepsilon _{11}^{(l) }\frac{v_\rho }{\sqrt{2}} &{} \varepsilon _{12}^{(l) } \frac{v_\rho }{\sqrt{2}} &{} \varepsilon _{13}^{(l) }\frac{v_\rho }{\sqrt{2}} \\ \varepsilon _{21}^{(l) }\frac{v_\rho }{\sqrt{2}}-x_E z_1 ^{(l) }\frac{v_\xi v_\rho }{\sqrt{2}m_E } &{} \varepsilon _{22}^{(l) }\frac{v_\rho }{\sqrt{2}}-x_E z_2 ^{(l) }\frac{v_\xi v_\rho }{\sqrt{2}m_E } &{} \varepsilon _{23}^{(l) }\frac{ v_\rho }{\sqrt{2}}-x_E z_3 ^{(l) }\frac{v_\xi v_\rho }{\sqrt{2}m_E } \\ \varepsilon _{31}^{(l) }\frac{v_\rho }{\sqrt{2}}+y_E x_1 ^{(l) }\frac{ v_{\varphi }v_\rho }{\sqrt{2}m_{E_3 }} &{} \varepsilon _{32}^{(l) }\frac{v_\rho }{\sqrt{2}}+y_E x_2 ^{(l) }\frac{v_{\varphi }v_\rho }{\sqrt{2}m_{E_3 }} &{} \varepsilon _{33}^{(l) }\frac{v_\rho }{\sqrt{2}}+y_E x_3 ^{(l) }\frac{ v_{\varphi }v_\rho }{\sqrt{2}m_{E_3 }} \end{array} \right) . \end{aligned}$$

(32)

In order to show that our model can naturally accommodate the experimental values of the charged lepton masses we use the extended benchmark scenario (26) assuming $m_{E} = m_{{\tilde{T}}}, m_{E_{3}}=m_{B_{3}}$. Then we have

$$\begin{aligned} v_\xi= & {} \lambda ^5\frac{vm_E }{v_\rho },\qquad v_{\varphi }=\lambda ^3 \frac{vm_{E_3 }}{v_\rho },\qquad \nonumber \\ \varepsilon _{ij}^{(l) }= & {} b_{ij}^{(l) }\lambda ^9\frac{v}{v_\rho },\qquad i,j=1,2,3. \end{aligned}$$

(33)

Here the one-loop contributions $\varepsilon ^{(l)}$ are estimated from their definitions in (31). Accordingly, the coefficients $b^{(l)}$ are constructed from the Yukawa and scalar quartic coupling. Thus, in the benchmark scenario (33) the SM charged lepton mass matrix reads:

$$\begin{aligned} \widetilde{M}_l= & {} \left( \begin{array}{ccc} b_{11}^{(l) }\lambda ^9 &{} b_{12}^{(l) }\lambda ^9 &{} b_{13}^{(l) }\lambda ^9 \\ b_{21}^{(l) }\lambda ^9+a_{21}^{(l) }\lambda ^5 &{} b_{22}^{(l) }\lambda ^9+a_{22}^{(l) }\lambda ^5 &{} b_{23}^{(l) }\lambda ^9+a_{23}^{(l) }\lambda ^5 \\ b_{31}^{(l) }\lambda ^9+a_{31}^{(l) }\lambda ^3 &{} b_{32}^{(l) }\lambda ^9+a_{32}^{(l) }\lambda ^3 &{} b_{33}^{(l) }\lambda ^9+a_{33}^{(l) }\lambda ^3 \end{array} \right) \nonumber \\ \end{aligned}$$

$$\begin{aligned} \frac{v}{\sqrt{2}}= & {} \left( \begin{array}{ccc} c_{11}^{(l) }\lambda ^9 &{} c_{12}^{(l) }\lambda ^9 &{} c_{13}^{(l) }\lambda ^9 \\ c_{21}^{(l) }\lambda ^5 &{} c_{22}^{(l) }\lambda ^5 &{} c_{23}^{(l) }\lambda ^5 \\ c_{31}^{(l) }\lambda ^3 &{} c_{32}^{(l) }\lambda ^3 &{} c_{33}^{(l) }\lambda ^3 \end{array} \right) \frac{v}{\sqrt{2}}, \end{aligned}$$

(34)

where

$$\begin{aligned} a^{(l)}_{21}&=-x_E z_1 ^{(l)},\qquad a^{(l)}_{22}=-x_E z_2 ^{(l) }, \qquad a^{(l)}_{23}=-x_E z_3 ^{(l)}, \end{aligned}$$

(35)

$$\begin{aligned} a^{(l)}_{31}&=y_E x_1 ^{(l) },\qquad \,\,\,\, a^{(l)}_{32}=y_E x_2 ^{(l) },\qquad \,\,\,\,\,\, a_{33}=y_E x_3 ^{\left( l\right) }, \end{aligned}$$

(36)

$$\begin{aligned} c_{21}^{(l) }&=b_{21}^{(l) }\lambda ^{4}+a_{21}^{(l) },\ \ c_{22}^{(l) }=b_{22}^{(l) }\lambda ^{4}+a_{22}^{(l) }, \ \ c_{23}^{(l) }=b_{23}^{(l) }\lambda ^{4}+a_{23}^{(l) }, \end{aligned}$$

(37)

$$\begin{aligned} c_{31}^{(l) }&=b_{31}^{(l) }\lambda ^{6}+a_{31}^{(l) },\ \ c_{32}^{(l) }=b_{32}^{(l) }\lambda ^{6}+a_{32}^{(l) },\ \ c_{33}^{(l) }=b_{33}^{(l) }\lambda ^{6}+a_{33}^{(l) }, \nonumber \\ c^{(l) }_{1i}&=b^{(l) }_{1i},\quad i=1,2,3. \end{aligned}$$

(38)

The matrix in the second equality of Eq. (34) is shown for convenience in order to explicitly display the hierarchy of the matrix elements of $\widetilde{M}_l$. To fit the measured values of the SM charged lepton masses [125], we solve the eigenvalue problem for the SM lepton mass matrix (34) and find the following solution:

$$\begin{aligned} c_{ij}^{(l)}=\left( \begin{array}{ccc} -1.13637 &{} -1.03665 &{} -0.866907 \\ -0.658689 &{} -0.525883 &{} 1.08155 \\ 0.900883 &{} -0.32514 &{} -0.312796 \\ &{} &{} \end{array} \right) \end{aligned}$$

(39)

An important point is that all the elements of this matrix constructed from Yukawa couplings are $\sim O (1)$. This means that the observed hierarchical charged lepton mass spectrum can be naturally reproduced in our model without significant tuning of the coupling constants.

6 Neutrino mass generation

The neutrino Yukawa interactions give rise to the following neutrino mass terms:

$$\begin{aligned} -{\mathcal {L}}_{\text {mass}}^{(\nu )}=\dfrac{1}{2} \begin{pmatrix} \overline{\nu _L^C}&\overline{\nu _R }&\overline{N_R } \end{pmatrix} M_{\nu } \begin{pmatrix} \nu _L \\ \nu _R ^C \\ N_R ^C \end{pmatrix} +H.c, \end{aligned}$$

(40)

where the neutrino mass matrix $M_{\nu }$ is

$$\begin{aligned} M_{\nu }= \begin{pmatrix} M_{1} &{} 0_{3\times 3} &{} 0_{3\times 3} \\ 0_{3\times 3} &{} M_{2} &{} M_{\chi } \\ 0_{3\times 3} &{} M_{\chi }^{T} &{} \mu \end{pmatrix} , \end{aligned}$$

(41)

with the submatrices $M_{1}$ and $M_{2}$ generated at one loop level, whereas the submatrices $M_{\chi }$ and $\mu$ appearing at tree level. They are given by:

$$\begin{aligned} M_{\chi }&=\left( \begin{array}{ccc} x_{N} &{} 0 &{} 0 \\ 0 &{} x_{N} &{} 0 \\ 0 &{} 0 &{} y_{N} \end{array} \right) \frac{v_{\chi }}{\sqrt{2}}, \nonumber \\ \mu&=\left( \begin{array}{ccc} h_{3N}v_{\sigma _{3}}-h_{1N}v_{\sigma _{1}} &{} h_{1N}v_{\sigma _{2}} &{} h_{2N}v_{\sigma _{2}} \\ h_{1N}v_{\sigma _{2}} &{} h_{3N}v_{\sigma _{3}}+h_{1N}v_{\sigma _{1}} &{} -h_{2N}v_{\sigma _{1}} \\ h_{2N}v_{\sigma _{2}} &{} -h_{2N}v_{\sigma _{1}} &{} h_{4N}v_{\sigma _{3}} \end{array} \right) . \end{aligned}$$

(42)

Fig. 5

One-loop Feynman diagram contributing to the entries of the light active neutrino mass matrix. Here $i,j,k,n=1,2,3$

The light active neutrino mass matrix is generated by the loop diagrams shown in Fig. 5 and is given by:

$$\begin{aligned} \widetilde{M}_{\nu }=M_{1}=\left( \begin{array}{ccc} x_{N}^{2}F\left( \mu _{22},m_{\chi _{1R}},m_{\chi _{1I}}\right) \mu _{22} &{} -x_{N}^{2}F\left( \mu _{12},m_{\chi _{1R}},m_{\chi _{1I}}\right) \mu _{12} &{} x_{N}y_{N}F\left( \mu _{23},m_{\chi _{1R}},m_{\chi _{1I}}\right) \mu _{23} \\ -x_{N}^{2}F\left( \mu _{12},m_{\chi _{1R}},m_{\chi _{1I}}\right) \mu _{12} &{} x_{N}^{2}F\left( \mu _{11},m_{\chi _{1R}},m_{\chi _{1I}}\right) \mu _{11} &{} -x_{N}y_{N}F\left( \mu _{13},m_{\chi _{1R}},m_{\chi _{1I}}\right) \mu _{13} \\ x_{N}y_{N}F\left( \mu _{23},m_{\chi _{1R}},m_{\chi _{1I}}\right) \mu _{23} &{} -x_{N}y_{N}F\left( \mu _{13},m_{\chi _{1R}},m_{\chi _{1I}}\right) \mu _{13} &{} y_{N}^{2}F\left( \mu _{33},m_{\chi _{1R}},m_{\chi _{1I}}\right) \mu _{33} \end{array} \right) , \end{aligned}$$

(43)

with the loop function of the form [126]:

$$\begin{aligned}&F\left( m_{1},m_{2},m_{3}\right) \nonumber \\&\quad =\frac{1}{16\pi ^{2}}\left[ \frac{m_{2}^{2} }{m_{2}^{2}-m_{1}^{2}}\ln \left( \frac{m_{2}^{2}}{m_{1}^{2}}\right) -\frac{ m_{3}^{2}}{m_{3}^{2}-m_{1}^{2}}\ln \left( \frac{m_{3}^{2}}{m_{1}^{2}}\right) \right] {\small .} \nonumber \\ \end{aligned}$$

(44)

In the limit where $\mu _{ij}^{2}\ll m_{\chi _{1R}}^{2}$, $m_{\chi _{1I}}^{2}$, the light active neutrino mass matrix becomes:

$$\begin{aligned} \widetilde{M}_{\nu }\simeq \frac{m_{\chi _{1R}}^{2}-m_{\chi _{1I}}^{2}}{8\pi ^{2}\left( m_{\chi _{1R}}^{2}+m_{\chi _{1I}}^{2}\right) }\left( \begin{array}{ccc} x_{N}^{2}\mu _{22} &{} x_{N}^{2}\mu _{12} &{} x_{N}y_{N}\mu _{23} \\ x_{N}^{2}\mu _{12} &{} x_{N}^{2}\mu _{11} &{} x_{N}y_{N}\mu _{13} \\ x_{N}y_{N}\mu _{23} &{} x_{N}y_{N}\mu _{13} &{} y_{N}^{2}\mu _{33} \end{array} \right) . \end{aligned}$$

(45)

All the elements of this mass matrix are free parameters and, therefore, our model does not predict specific values of neutrino masses and mixing. However, in our model, the small value of the overall neutrino mass scale is natural. As seen from Eq. (45), the smallness of this scale is attributed to a small splitting $\Delta m_{\chi }^{2}$ between the masses of the $\chi _{1R}$ and $\chi _{1I}$ scalar fields, which originates from the quartic term $\gamma \left( \chi ^{\dagger }\eta \sigma _{3}\varphi +h.c\right)$, so that $\Delta m_{\chi }^{2}\sim \gamma v_{\varphi }^{2}$. (see Appendix B). Requiring smallness of the parameter $\kappa$, we must guarantee its stability with respect radiative corrections, i.e. its technical naturalness. Checking the model Lagrangian, we observe that in the limit $\gamma \rightarrow 0$ it acquires an extra symmetry, protecting this parameter from large radiative corrections. Here we do not need to specify this group completely and just give its minimal non-trivial subgroup. This is $Z_{3}$ with the field assignment, where all leptonic fields as well as the scalar fields $\sigma$ and $\sigma _{3}$ have a charge equal to $\omega =e^{\frac{2\pi i}{3}}$, whereas the remaining fields are neutral under this symmetry. This symmetry is broken by the coupling $\gamma$. Therefore, in our model small masses of the light neutrinos are technically natural, being protected by this accidental symmetry. As a result, the components $\chi _{1,2}$ of the scalar $SU(3)_{L}$-triplet can be sufficiently light to provide a non-trivial phenomenology.

7 Conclusions

We have constructed a renormalizable theory based on the $SU(3)_C\times SU(3)_L\times U(1)_X$ gauge symmetry, supplemented with the spontaneously broken $U(1)_{L_g}$ global lepton number symmetry and the $S_3 \times Z_2$ discrete group, consistent with the low energy SM fermion flavor data. In our model, the particle spectrum of the 3-3-1 model with right handed Majorana neutrinos is enlarged by the inclusion of gauge singlet scalars and charged exotic vector like fermions, which are crucial for the implementation of the tree level Universal seesaw mechanism that produces the masses for the bottom, strange and charm quarks as well as the tau and muon lepton masses. The top and exotic quarks obtain their tree level masses from renormalizable Yukawa interactions, whereas the first generation SM charged fermion masses are generated from a one loop level radiative seesaw mechanism. The masses for the light active neutrinos arise from a radiative seesaw mechanism at one loop level. The natural smallness of the overall neutrino mass scale is guarantied by an accidental softly broken symmetry. Our model successfully explains the hierarchy of the fermion masses and mixings as well as accommodates the current experimental deviations of the electron and muon anomalous magnetic moments from their SM values.

Appendices

Appendix A: The $S_3$ discrete group

The $S_3$ discrete group contains 3 irreducible representations: ${\mathbf {1}}$, ${\mathbf {1}}^{\prime }$ and ${\mathbf {2}}$. Considering $\left( x_1 ,x_2 \right) ^T$ and $\left( y_1 ,y_2 \right) ^T$ as the basis vectors for two $S_3$ doublets and $(y{\acute{}})$ is an $S_3$ non trivial singlet, the multiplication rules of the $S_3$ group for the case of real representations take the form [127]:

$$\begin{aligned} \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) _{{\mathbf {2}}}\otimes \left( \begin{array}{c} y_1 \\ y_{2} \end{array} \right) _{{\mathbf {2}}}= & {} \left( x_1 y_1 +x_2 y_2 \right) _{{\mathbf {1}}}+\left( x_1 y_2 -x_2 y_1 \right) _{{\mathbf {1}}^{\prime }} \nonumber \\&+\left( \begin{array}{c} x_2 y_2 -x_1 y_1 \\ x_1 y_2 +x_2 y_1 \end{array} \right) _{{\mathbf {2}}}, \end{aligned}$$

(A1)

$$\begin{aligned} \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) _{{\mathbf {2}}}\otimes \left( y {\acute{}} \right) _{{\mathbf {1}}^{\prime }}= & {} \left( \begin{array}{c} -x_2 y {\acute{}} \\ x_1 y {\acute{}} \end{array} \right) _{{\mathbf {2}}},\qquad \nonumber \\ \left( x {\acute{}} \right) _{{\mathbf {1}}^{\prime }}\otimes \left( y {\acute{}} \right) _{{\mathbf {1}}^{\prime }}= & {} \left( x {\acute{}} y {\acute{}} \right) _{{\mathbf {1}}}. \end{aligned}$$

(A2)

Appendix B: The scalar potential

The renormalizable scalar potential of our model takes the form:

$$\begin{aligned} V= & {} -\mu _{\chi }^{2}(\chi ^{\dagger }\chi )-\mu _{\eta }^{2}(\eta ^{\dagger }\eta )-\mu _{\rho }^{2}(\rho ^{\dagger }\rho )-\mu _{\xi }^{2}(\xi \xi )_{{\mathbf {1}}}\nonumber \\&-\mu _{\varphi }^{2}\varphi ^{2}-\mu _{\sigma }^{2}(\sigma ^{\dagger }\sigma )_{{\mathbf {1}}}-\mu _{\sigma _{3}}^{2}(\sigma _{3}^{\dagger }\sigma _{3}) \nonumber \\&+\left( \kappa \varphi \eta _{i}\chi _{j}\rho _{k}\varepsilon ^{ijk}+H.c.\right) +\gamma \left( \chi ^{\dagger }\eta \sigma _{3}\varphi +h.c\right) \nonumber \\&+\lambda _{1}(\chi ^{\dagger }\chi )(\chi ^{\dagger }\chi )+\lambda _{2}(\rho ^{\dagger }\rho )(\rho ^{\dagger }\rho ) +\lambda _{3}(\eta ^{\dagger }\eta )(\eta ^{\dagger }\eta ) \nonumber \\&+\lambda _{4}(\chi ^{\dagger }\chi )(\rho ^{\dagger }\rho )+\lambda _{5}(\chi ^{\dagger }\chi )(\eta ^{\dagger }\eta )+\lambda _{6}(\rho ^{\dagger }\rho )(\eta ^{\dagger }\eta )\nonumber \\&+\lambda _{7}(\chi ^{\dagger }\eta )(\eta ^{\dagger }\chi )+\lambda _{8}(\chi ^{\dagger }\rho )(\rho ^{\dagger }\chi )+\lambda _{9}(\rho ^{\dagger }\eta )(\eta ^{\dagger }\rho ) \nonumber \\&+\lambda _{10}\left[ (\xi \xi )_{{\mathbf {1}}}\right] ^{2}+\lambda _{11} \left[ (\xi \xi )_{{\mathbf {1}}^{\prime }}\right] ^{2}+\lambda _{12}\left[ (\xi \xi )_{{\mathbf {2}}}(\xi \xi )_{{\mathbf {2}}}\right] _{{\mathbf {1}}}\nonumber \\&+\lambda _{13}\varphi ^{4}+\lambda _{14}\left[ (\sigma ^{\dagger }\sigma )_{{\mathbf {1}} }\right] ^{2}+\lambda _{15}\left[ (\sigma ^{\dagger }\sigma )_{{\mathbf {1}} ^{\prime }}\right] ^{2} \nonumber \\&+\lambda _{16}\left[ (\sigma ^{\dagger }\sigma )_{ {\mathbf {2}}}\right] ^{2} \nonumber \\&+\lambda _{17}(\sigma _{3}^{\dagger }\sigma _{3})^{2}+\lambda _{18}(\xi \xi )_{{\mathbf {1}}}\varphi ^{2}+\lambda _{19}\left[ (\xi \xi )_{{\mathbf {1}}} \right] (\sigma ^{\dagger }\sigma )_{{\mathbf {1}}}\nonumber \\&+\lambda _{20}\left[ (\xi \xi )_{{\mathbf {1}}^{\prime }}\right] (\sigma ^{\dagger }\sigma )_{{\mathbf {1}} ^{\prime }}+\lambda _{21}\left[ (\xi \xi )_{{\mathbf {2}}}\right] (\sigma ^{\dagger }\sigma )_{{\mathbf {2}}} \nonumber \\&+\lambda _{22}(\xi \xi )_{{\mathbf {1}}}(\sigma _{3}^{\dagger }\sigma _{3})+\lambda _{23}\varphi ^{2}(\sigma ^{\dagger }\sigma )_{{\mathbf {1}} }+\lambda _{24}(\sigma ^{\dagger }\sigma )_{{\mathbf {1}}}(\sigma _{3}^{\dagger }\sigma _{3})\nonumber \\&+\lambda _{25}\left[ (\sigma \sigma )_{{\mathbf {1}}}(\sigma _{3}^{\dagger }\sigma _{3}^{\dagger })+h.c\right] + \lambda _{26}\varphi ^{2}(\chi ^{\dagger }\chi ) \nonumber \\&+\lambda _{27}(\xi \xi )_{{\mathbf {1}}}(\chi ^{\dagger }\chi )+\lambda _{28}(\sigma ^{\dagger }\sigma )_{{\mathbf {1}}}(\chi ^{\dagger }\chi )\nonumber \\&+\lambda _{29}(\sigma _{3}^{\dagger }\sigma _{3})(\chi ^{\dagger }\chi )+\lambda _{30}(\xi \xi )_{{\mathbf {1}}}(\eta ^{\dagger }\eta )+\lambda _{31}\varphi ^{2}(\eta ^{\dagger }\eta ) \nonumber \\&+\lambda _{32}(\sigma ^{\dagger }\sigma )_{{\mathbf {1}}}(\eta ^{\dagger }\eta )+\lambda _{33}(\sigma _{3}^{\dagger }\sigma _{3})(\eta ^{\dagger }\eta )\nonumber \\&+\lambda _{34}(\xi \xi )_{{\mathbf {1}}}(\rho ^{\dagger }\rho )+\lambda _{35}\varphi ^{2}(\rho ^{\dagger }\rho )+\lambda _{36}(\sigma ^{\dagger }\sigma )_{{\mathbf {1}}}(\rho ^{\dagger }\rho ) \nonumber \\&+\lambda _{37}(\sigma _{3}^{\dagger }\sigma _{3})(\rho ^{\dagger }\rho )+\lambda _{38}\left\{ \left[ \left( \xi \xi \right) _{{\mathbf {2}}}\xi \right] _{{\mathbf {1}}^{\prime }}\varphi +h.c\right\} \nonumber \\&+\lambda _{39}\left[ (\xi \xi )_{ {\mathbf {2}}}(\sigma ^{\dagger }\sigma _{3})+h.c\right] \nonumber \\&+\lambda _{40}\left[ (\sigma ^{\dagger }\xi )_{{\mathbf {1}}^{\prime }}\varphi \sigma _{3}+h.c\right] \end{aligned}$$

(B1)

where $\lambda _{i}$ ($i=1,2,\ldots ,40$) are dimensionless parameters, whereas $\mu _{\chi }$, $\mu _{\eta }$, $\mu _{\xi }$, $\mu _{\varphi }$, $\mu _{\sigma }$, $\mu _{\sigma _{3}}$, f and A are dimensionful parameters. Here $\chi$, $\rho$ and $\eta$, the $SU(3)_{L}$ scalar triplets and the remaining fields are $SU(3)_{L}$ scalar singlets. The scalar fields $\sigma$ and $\sigma _{3}$ are complex, wheras $\xi$ and $\varphi$ are real.

Appendix C: The scalar potential and scalar mass matrix entries for the g-2 contribution

The scalar potential that contributes to $g-2$ anomalies is given by:

$$\begin{aligned} V^{\text {CP}}= & {} -\mu _{\rho }^{2}(\rho ^{\dagger }\rho )-\mu _{\xi }^{2}(\xi \xi )_{{\mathbf {1}}}-\mu _{\varphi }^{2}\varphi ^{2} \nonumber \\&+\left( \kappa \varphi \eta _{i}\chi _{j}\rho _{k}\varepsilon ^{ijk}+H.c.\right) \nonumber \\&+\lambda _{2}(\rho ^{\dagger }\rho )(\rho ^{\dagger }\rho )+\lambda _{10} \left[ (\xi \xi )_{{\mathbf {1}}}\right] ^{2}+\lambda _{11}\left[ (\xi \xi )_{ {\mathbf {1}}^{\prime }}\right] ^{2} \nonumber \\&+\lambda _{12}\left[ (\xi \xi )_{{\mathbf {2}}}(\xi \xi )_{{\mathbf {2}}}\right] _{{\mathbf {1}}}+\lambda _{13}\varphi ^{4}+\lambda _{18}(\xi \xi )_{{\mathbf {1}} }\varphi ^{2} \nonumber \\&+\lambda _{34}(\xi \xi )_{{\mathbf {1}}}(\rho ^{\dagger }\rho )+\lambda _{35}\varphi ^{2}(\rho ^{\dagger }\rho ) \nonumber \\&+\lambda _{38}\left\{ \left[ \left( \xi \xi \right) _{{\mathbf {2}}}\xi \right] _{{\mathbf {1}}^{\prime }}\varphi +h.c\right\} \end{aligned}$$

(C1)

with the complex quartic coupling $\kappa =\kappa _{r}+i\kappa _{i}$ introducing CP-violation in the scalar potential. We need it in order to mix $\varphi$ with $\zeta _{\rho }$. The scalar potential minimization equations allow us to express the $\mu _{\rho }$, $\mu _{\xi }$ and $\mu _{\varphi }$ parameters as follows:

$$\begin{aligned} \mu _{\rho }^{2}&=\frac{1}{2}\left( \frac{{\kappa _{r}}v_{\eta }v_{\varphi }v_{\chi }}{v_{\rho }}+\lambda _{34}v_{\xi }^{2}+2\lambda _{2}v_{\rho }^{2}+\lambda _{35}v_{\varphi }^{2}\right) \end{aligned}$$

(C2)

$$\begin{aligned} \mu _{\xi }^{2}&=\frac{1}{2}\left( 2\left( \lambda _{10}+\lambda _{12}\right) v_{\xi }^{2}+\lambda _{34}v_{\rho }^{2}+\lambda _{18}v_{\varphi }^{2}\right) \end{aligned}$$

(C3)

$$\begin{aligned} \mu _{\varphi }^{2}&=\frac{1}{2}\left( \frac{{\kappa _{r}}v_{\eta }v_{\rho }v_{\chi }}{v_{\varphi }}+\lambda _{18}v_{\xi }^{2}+\lambda _{35}v_{\rho }^{2}+2\lambda _{13}v_{\varphi }^{2}\right) \end{aligned}$$

(C4)

The potential (C1) generates the entries of the scalar mass squared matrix ${\mathbf {M}}^{2}$. In the basis $\xi _{\rho }$ , $\xi _{1R}$, $\xi _{2R}$, $\varphi$, $\zeta _{\rho }$ these entries are given by:

$$\begin{aligned} m_{11}^{2}&=2\lambda _{2}v_{\rho }^{2}-\frac{{\kappa _{r}}v_{\eta }v_{\varphi }v_{\chi }}{2v_{\rho }} \end{aligned}$$

(C5)

$$\begin{aligned} m_{12}^{2}&=m_{21}^{2}=\lambda _{34}v_{\xi }v_{\rho } \end{aligned}$$

(C6)

$$\begin{aligned} m_{14}^{2}&=m_{41}^{2}=\frac{1}{2}{\kappa _{r}}v_{\eta }v_{\chi }+\lambda _{35}v_{\rho }v_{\varphi } \end{aligned}$$

(C7)

$$\begin{aligned} m_{22}^{2}&=2\left( \lambda _{10}+\lambda _{12}\right) v_{\xi }^{2} \end{aligned}$$

(C8)

$$\begin{aligned} m_{23}^{2}&=m_{32}^{2}=-2\sqrt{2}\lambda _{38}v_{\xi }v_{\varphi } \end{aligned}$$

(C9)

$$\begin{aligned} m_{24}^{2}&=m_{42}^{2}=\lambda _{18}v_{\xi }v_{\varphi } \end{aligned}$$

(C10)

$$\begin{aligned} m_{33}^{2}&=2\lambda _{10}v_{\xi }^{2}-3\lambda _{12}v_{\xi }^{2}-2\lambda _{38}v_{\varphi }v_{\xi } \end{aligned}$$

(C11)

$$\begin{aligned} m_{34}^{2}&=m_{43}^{2}=-\sqrt{2}\lambda _{38}v_{\xi }^{2} \end{aligned}$$

(C12)

$$\begin{aligned} m_{44}^{2}&=2\lambda _{13}v_{\varphi }^{2}-\frac{{\kappa _{r}} v_{\eta }v_{\rho }v_{\chi }}{2v_{\varphi }} \end{aligned}$$

(C13)

$$\begin{aligned} m_{45}^{2}&=m_{54}^{2}=-\frac{1}{2}{\kappa _{i}}v_{\eta }v_{\chi } \end{aligned}$$

(C14)

$$\begin{aligned} m_{55}^{2}&=-\frac{{\kappa _{r}}v_{\eta }v_{\varphi }v_{\chi }}{ 2v_{\rho }}. \end{aligned}$$

(C15)

From these expressions one can see that by the appropriate choice of the signs of the quartic couplings it easy to guarantee positive definiteness of the mass squared matrix (12) and, therefore, that the extremum conditions (C2)–(C4) correspond to the minimum of the potential (C1).