Exploring a Gauge Horizontal Model for Charged Fermion Masses

Upon careful examination of the lepton-related interaction terms in the Lagrangian, as detailed in Eqs. (14) and (29), it becomes evident that the model exhibits three accidental global $U(1)$ lepton flavor symmetries. Specifically, $L_{1}$, $e_{R}$, and $N_{L,R}$ each carry one unit of electron number, while $\{L_{2},\mu_{R}\}$ and $\{L_{3},\tau_{R}\}$ carry one unit of muon and tau number, respectively. All fields other than leptons and $N_{L,R}$ are neutral under these lepton numbers. These global $U(1)$ lepton symmetries are preserved even after electroweak symmetry breaking, leading to a see-saw plus FN suppressed electron neutrino mass, $m_{\nu_{e}}\sim(y^{e}_{3,eff}v_{0})^{2}/M_{N}$¹¹1We estimate a sub-eV range for $m_{\nu_{e}}$ by applying the FN suppression factor for the $H_{3}$ Yukawa, as discussed in the next section. Assuming $M_{N}\sim\text{PeV}$ and $y_{e}^{3}\sim 0.1$, we obtain this ballpark value. , massless $\nu_{\mu,\tau}$, and ensuring compliance with stringent limits on charged lepton flavor violation.

In contrast to the quark sector, the $U(1)$ lepton flavor symmetry in this model preserves the structural zeros in the charged lepton mass matrix, preventing them from being lifted by quantum corrections.

To simplify the discussion and reduce the number of free parameters, we consider a scenario where only $y^{u}_{+}$ is complex. This assumption effectively eliminates CP violation in kaon mixing and addresses the CKM weak CP phase. For illustration, consider the following benchmark point:

	$\displaystyle M_{u}$	$\displaystyle\simeq$	$\displaystyle\left(\begin{array}[]{ccc}4.71\times 10^{-4}&0.0888&0.0407\\ 2.80\times 10^{-4}&0.392&0.331\,e^{0.437i}\\ 1.078&2.25\times 10^{-3}&114.67\\ \end{array}\right)\,\text{GeV}\,,$		(36)
	$\displaystyle M_{d}$	$\displaystyle\simeq$	$\displaystyle\left(\begin{array}[]{ccc}1.59\times 10^{-3}&1.78\times 10^{-4}&0.016\\ 1.05\times 10^{-4}&0.0361&0.0796\\ -4.549\times 10^{-3}&9.59\times 10^{-4}&1.722\\ \end{array}\right)\,\text{GeV}\,.$		(40)

These quark mass matrices can be generated by the following set of model parameters:

	$\displaystyle v_{1}$	$\displaystyle=$	$\displaystyle 2\,\mathrm{PeV}\,,\quad M_{+}=26.45\,\mathrm{PeV}\,,\quad M_{-}=22.75\,\mathrm{PeV}\,,\quad M_{3}=12.48\,\mathrm{PeV}\,,\quad M_{1}=22.24\,\mathrm{PeV}\,,$
	$\displaystyle\lambda_{-,0,s,s}$	$\displaystyle=$	$\displaystyle 0.527\,,\quad\lambda_{+,0,s,s}=0.696\,,$
	$\displaystyle\mu_{0,1,s}$	$\displaystyle=$	$\displaystyle 2.97\,\mathrm{PeV}\,,\quad\mu_{-,3,s}=11.22\,\mathrm{PeV}\,,\quad(\lambda_{+,-,0,0}\times\mu_{+,1,s})\simeq 4.56\,\mathrm{PeV}\,,$
	$\displaystyle y^{u}_{3}$	$\displaystyle=$	$\displaystyle 0.870\,,\quad y^{u}_{+}=0.477e^{0.437i}\,,\quad y^{u}_{-}=0.057\,,\quad y^{u}_{1}=0.187\,,\quad y^{\prime u}_{1}=0.515\,,$
	$\displaystyle y^{d}_{3}$	$\displaystyle=$	$\displaystyle-0.0445\,,\quad y^{d}_{+}=0.0233\,,\quad y^{d}_{-}=0.112\,,\quad y^{d}_{1}=0.0172\,,\quad y^{e}_{1}=0.0502\,.$		(41)

Observe that $\delta m_{u}$ cannot constrain the two individual parameters but only their product, $\lambda_{+,-,0,0}\times\mu_{+,1,s}$. Note that the absolute values of the Yukawa coupling strengths to the exotic Higgs doublets are approximately in the ranges of $[0.1,1.0]$ for the up-quark sector and $[0.01,0.2]$ for the down-quark and charged lepton sectors. This substantial reduction significantly narrows the five-order-of-magnitude differences observed in the SM Yukawas. Notably, the muon Yukawa coupling to $H_{1}$ exceeds the SM tau Yukawa coupling, with $y^{e}_{1}=0.0502$ significantly larger than $y^{\tau}_{\text{SM}}=0.0072$. Similarly, the Yukawa coupling for the strange quark, $y^{d}_{1}$, surpasses the SM bottom quark Yukawa coupling, $y^{b}_{\text{SM}}$.

Regarding quantum corrections, we focus on the $(1,1)$ components of the two mass matrices. Equations (28) and (27) are used for estimation, as they provide the dominant contributions to the up and down quark masses. Quantum corrections to other tree-level structure zeros are assumed to be controlled by additional model parameters and are not expected to have significant effects.

This approach results in a fit that aligns well with the previously mentioned values listed in Eq. (31) and Eq. (32). It is noteworthy that the mass matrix elements associated with the FN mechanism require only an order-one tuning in their Yukawa couplings. Additionally, the elements that arise through radiative processes are generally small and exhibit a hierarchy of less than order one among themselves.

Using the benchmark values mentioned above, we have determined the scalar FN suppression factors for the new Higgs doublets relative to the SM Yukawa couplings. The hierarchical suppression factors, see Eq.(22), are $\frac{v_{1}\mu_{0,1,s}}{M_{1}^{2}}=0.012$ for $H_{1}$, $\lambda_{\pm,0,s,s}\times\frac{v_{1}^{2}}{M_{\pm}^{2}}\simeq 0.004$ for $H_{\pm}$, and $\lambda_{-,0,s,s}\times\frac{v_{1}^{3}\mu_{-,3,s}}{M_{-}^{2}M_{3}^{2}}=0.00059$ for $H_{3}$, respectively.

One must account for the stringent constraints imposed by Flavor-Changing Neutral Currents (FCNCs) mediated by the new gauge boson. In the mass basis, the couplings to the new gauge boson are given by:

$$Q^{q}_{L/R}=(V^{q}_{L/R})^{\dagger}\cdot\text{diag}\{q^{q}_{L/R},-q^{q}_{L/R},0\}\cdot V^{q}_{L/R}\,,$$

(42)

where $V^{q}_{L/R}$ are the mixing matrices that diagonalize the quark mass matrix.

Using the benchmark solution, the gauge couplings in the quark mass basis at $\mu=M_{F}$ are as follows:

	$\displaystyle Q^{d}_{L}$	$\displaystyle\simeq$	$\displaystyle\left(\begin{array}[]{ccc}1.999&-0.0195&0.0192\\ -0.0195&-1.999&0.0921\\ 0.0192&0.0921&-0.00409\\ \end{array}\right)\,,$		(46)
	$\displaystyle Q^{d}_{R}$	$\displaystyle\simeq$	$\displaystyle\left(\begin{array}[]{ccc}2.999&0.0537&-0.00783\\ 0.0537&-2.999&-0.0046\\ -0.00783&-0.0046&1.37\times 10^{-5}\\ \end{array}\right)\,,$		(50)
	$\displaystyle Q^{u}_{L}$	$\displaystyle\simeq$	$\displaystyle\left(\begin{array}[]{ccc}1.805&0.861\,e^{-0.44i}&0.0019\,e^{2.56i}\\ 0.862\,e^{0.44i}&-1.805&0.0055\,e^{0.023i}\\ 0.0019\,e^{-2.56i}&0.0055\,e^{-0.023i}&-1.63\times 10^{-5}\\ \end{array}\right)\,,$		(54)
	$\displaystyle Q^{u}_{R}$	$\displaystyle\simeq$	$\displaystyle\left(\begin{array}[]{ccc}0.999&0.0137\,e^{0.253i}&-0.0094\\ 0.0137\,e^{-0.253i}&-0.999&9.3\times 10^{-5}\,e^{2.78i}\\ -0.0094&9.3\times 10^{-5}\,e^{-2.78i}&8.84\times 10^{-5}\\ \end{array}\right)\,.$		(58)

The most stringent experimental constraints arise from $K^{0}-\bar{K^{0}}$, $B^{0}-\bar{B^{0}}$, and $D^{0}-\bar{D^{0}}$ mixings. Consequently, our primary focus is on the relevant 4-fermi operators. In conventional notation, as detailed in [31], the only non-zero Wilson coefficients at $M_{F}$ are associated with the flavor-crossing gauge couplings are:

$$C_{ij}^{1}={[(Q_{L})_{ij}]^{2}\over v_{1}^{2}}\,,\widetilde{C}_{ij}^{1}={[(Q_{R})_{ij}]^{2}\over v_{1}^{2}}\,,C_{ij}^{5}=-4{(Q_{L})_{ij}(Q_{R})_{ij}\over v_{1}^{2}}\,,$$

(59)

These coefficients are independent of the coupling constant since $M_{F}=g_{F}v_{1}$, and only the VEV $v_{1}$ is relevant.

However, at the corresponding meson scale, non-vanishing $C_{ij}^{4}$ emerges from the RGE running down from $M_{F}$ [31, 22]. In this benchmark, the Wilson coefficients for $K^{0}\mathchar 45\relax\overline{K^{0}}$ mixing at $\mu=M_{F}$ are given by:

$$C^{1}_{K}(M_{F})=3.81\times 10^{-4}v_{1}^{-2}\,,\;\widetilde{C}^{1}_{K}(M_{F})=2.89\times 10^{-3}v_{1}^{-2}\,,\;C^{5}_{K}(M_{F})=4.19\times 10^{-3}v_{1}^{-2}\,.$$

(60)

The RGE running yields:

	$\displaystyle C^{1}_{K}(\mu_{K})=2.51\times 10^{-4}v_{1}^{-2}\,,\;\widetilde{C}^{1}_{K}(\mu_{K})=1.89\times 10^{-3}v_{1}^{-2}\,,$
	$\displaystyle C^{5}_{K}(\mu_{K})=3.65\times 10^{-3}v_{1}^{-2}\,,\;C^{4}_{K}(\mu_{K})=1.18\times 10^{-2}v_{1}^{-2}\,,$		(61)

where $\mu_{K}=2\,\text{GeV}$.

We have established that the most stringent constraint is $|\Re C^{4}_{K}|<3.6\times 10^{-15}\ \text{GeV}^{-2}$ [32], corresponding to $v_{1}>1.81\ \text{PeV}$. Given the sizable $(12)$ component of $|Q^{u}_{L}|$ in this benchmark, we carefully verified that, even after considering the RG running [33, 32] to $\mu_{D}=2.8\ \text{GeV}$, the constraint $|C^{4}_{D}(\mu_{D})|=0.109v_{1}^{-2}\ \text{GeV}^{-2}<4.8\times 10^{-14}\ \text{GeV}^{-2}$ [32] still imposes a slightly weaker bound of $v_{1}>1.51\ \text{PeV}$ compared to $\Re C^{4}_{K}$. Therefore, it is permissible to select $M_{F}=1.0\ \text{PeV}$ with $v_{1}=2\ \text{PeV}$ and $g_{F}=0.5$.

Nevertheless, the new gauge coupling strength is an unknown parameter. If one assumes a smaller $g_{F}$, the mass of the new gauge boson is lighter. For example, $M_{F}\simeq 3\ \text{TeV}$ if $g_{F}\simeq 0.0015$, and it may be probed in a future high-energy $e^{-}e^{+}$ collider [34, 35, 36]. For more discussion, see Section IV.1.

To gain deeper insights into the behavior of the model, we extend our previous analysis by conducting a thorough numerical scan of the model parameters. This comprehensive study enables us to assess how naturally the model accommodates realistic data.

We adopt the benchmark values for $v_{1}$, $M_{\pm}$, $M_{3}$, and $M_{1}$, along with the corresponding scalar FN suppression factors derived from Eq. (41). For the $(33)$ components, we keep $y_{t}$ and $y_{b}$ within $10\%$ of their SM values. For other Yukawa couplings, we allow the absolute values to vary between $0.1$ ($0.01$) and $1.0$ ($0.1$) for the up (down) quark sector, each with an arbitrary complex phase. Each tree-level structural zero element is constrained to have an absolute value of less than $2\ \mathrm{MeV}$ with a random phase.

Approximately $2\%$ of the randomly generated configurations exhibit a mild hierarchical structure, characterized by $0.3>|V_{us}|>3|V_{cb}|>15|V_{ub}|$, $m_{b}>5m_{s}>25m_{d}$, and $m_{t}>5m_{c}>25m_{u}$. Statistics of the resulting CKM elements, $C_{K}^{4}$, and $J$ from 5000 such configurations are displayed in Fig. 4 and Fig. 5.

From Fig. 4, it is evident that the observed CKM mixing pattern falls within the expected range of this model, requiring minimal fine-tuning. Additionally, the absolute value of $J$ is approximately correct, as illustrated in the right panel of Fig. 5.

Regarding the low-energy $\Delta S=2$ FCNC constraint, this model provides ample parameter space to accommodate realistic configurations, as illustrated in Fig. 5. Notably, selecting $v_{1}=10$ PeV instead of $2$ PeV shifts the central value of $|\text{Im}(C^{4}_{K})|$ from approximately $10^{-16.4}\text{GeV}^{-2}$ to $10^{-17.8}\text{GeV}^{-2}$ (see Fig. 6), thereby easily circumventing the constraint $\text{Im}(C^{4}_{K})\in[-1.8,0.9]\times 10^{-17}\text{GeV}^{-2}$ [32].

Due to the gauge nature of the model, the existence of a new gauge boson, $Z_{F}$, is inevitable, and its discovery would serve as compelling evidence. The mass of $Z_{F}$ is an unknown parameter. If the theory remains perturbative, one can only infer that $M_{F}\lesssim v_{1}$. Note, however, that the low-energy FCNC four-fermion effective operators, mediated by tree-level flavor-crossing $Z_{F}$ couplings, are determined solely by the $U(1)_{F}$ charges and $v_{1}$, and are independent of $M_{F}$.

The value of $v_{1}$ remains unknown. If $v_{1}$ is relatively low, we might have a chance to detect the footprint of $Z_{F}$ with improved low-energy FCNC experimental data. For instance, if $v_{1}$ is $2\,\text{PeV}$, as shown in the left panel of Fig. 6, the ‘natural’ region for $|C_{D}^{4}|$ is $\gtrsim 10^{-15}\mathrm{GeV}^{-2}$ and for $|\text{Im}\,C_{K}^{4}|$ is $\gtrsim 10^{-18}\mathrm{GeV}^{-2}$, thus an order of magnitude improvement in experimental sensitivity could be sufficient. However, if $v_{1}$ is pushed to $10\,\text{PeV}$, as shown in the right panel of Fig. 6, the lower bound of the ‘natural’ region for $|C_{D}^{4}|$ shifts to $\gtrsim 10^{-16}\mathrm{GeV}^{-2}$ and for $|\text{Im}\,C_{K}^{4}|$ to $\gtrsim 10^{-19}\mathrm{GeV}^{-2}$.

Next, we will discuss the phenomenologically interesting scenario where $Z_{F}$ is relatively light, specifically $M_{F}\lesssim 10\,\text{TeV}$, such that $M_{1,3,\pm,N}\gg M_{F}$. In addition to contributing to FCNC $\Delta F=2$ transitions, $Z_{F}$ might also exhibit interesting signatures at colliders in the near future.

If the SM fermion masses can be neglected, one can transition to the interaction basis to calculate the total decay width, which is proportional to $\sum_{f}\left[(Q_{L}^{f})^{2}+(Q_{R}^{f})^{2}\right]$. Using the $U(1)_{F}$ charges from Table 1, the decay width of $Z_{F}$ can be expressed as:

$$\Gamma_{F}=\frac{151}{12\pi}M_{F}g_{F}^{2}=4.01\frac{M_{F}^{3}}{v_{1}^{2}}=8.02\left(\frac{M_{F}}{20\,\text{TeV}}\right)^{3}\left(\frac{2\,\text{PeV}}{v_{1}}\right)^{2}\,\text{GeV}.$$

(62)

Comparing this to its mass,

$$\frac{\Gamma_{F}}{M_{F}}=4.01\frac{M_{F}^{2}}{v_{1}^{2}}=4.01\times 10^{-6}\left(\frac{M_{F}}{2\,\text{TeV}}\right)^{2}\left(\frac{2\,\text{PeV}}{v_{1}}\right)^{2}\,,$$

(63)

indicating an extremely narrow resonance when the gauge boson is relatively light and directly produced at colliders.

It is essential to measure the coupling strengths of the SM fermions to distinguish this model from other $Z^{\prime}$ models. Notably, by construction, there are almost no flavor-diagonal couplings to $\tau$, $\nu_{\tau}$, top quark, and bottom quark. For the first and second generations, the actual flavor-dependent couplings depend on the specifics of the quark mass matrix diagonalization, and hence, there is no definitive prediction. However, we anticipate that the gross pattern should not vary significantly. For illustration, using the benchmark configuration discussed earlier, this model predicts the following ratio for the production cross-sections near the resonance of $Z_{F}$ in different channels:

	$\displaystyle\sigma(e^{+}e^{-}\to Z_{F}^{*}\to e\bar{e}):\sigma(e^{+}e^{-}\to Z_{F}^{*}\to\mu\bar{\mu}):\sigma(e^{+}e^{-}\to Z_{F}^{*}\to s\bar{s}):\sigma(e^{+}e^{-}\to Z_{F}^{*}\to d\bar{d})$
	$\displaystyle:\sigma(e^{+}e^{-}\to Z_{F}^{*}\to u\bar{u}):\sigma(e^{+}e^{-}\to Z_{F}^{*}\to c\bar{c})\simeq 4.8:4.8:3.1:3.1:1:1$		(64)

with

$$\sigma(e^{+}e^{-}\to Z_{F}^{*}\to\mu\bar{\mu})\simeq\left(\frac{61}{151}\right)^{2}\frac{6\pi}{M_{F}^{2}}=1.19\times\left(\frac{\mathrm{TeV}}{M_{F}}\right)^{2}\,\mbox{nb}$$

(65)

Moreover, the forward-backward asymmetries of electrons and muons are predicted to deviate significantly from the SM predictions near the $Z_{F}$ resonance, whereas the $\tau$ lepton does not exhibit such modifications, as shown in Fig. 7. Since the first- and second-generation charged leptons possess opposite $U(1)_{F}$ charges, their asymmetry lineshapes form a mirror image with respect to the resonance. Once observed, this phenomenon would provide a convincing indication of the existence of this specific gauged $U(1)$ symmetry.

Before closing this subsection, we remark that although only the SM Higgs and $S_{1}$ develop nonzero VEVs, the SM Higgs can still mix with $H_{1}$ through the cubic coupling $\mu_{0,1,s}\overline{H_{1}}HS_{1}$. Using the benchmark point, the mixing angle is determined by

$$\theta_{H}=\frac{1}{2}\tan^{-1}\left(\frac{\mu_{0,1,s}v_{1}}{M_{1}^{2}-M_{h}^{2}}\right)=0.006\,,$$

(66)

where $M_{h}$ is the mass of the SM Higgs. This results in a universal suppression factor, $\cos\theta_{H}$, applied to all SM Higgs couplings, leading to an ${\cal O}(10^{-5})$ deviation from the SM prediction. Such a small deviation poses significant experimental challenges for detection.

Here, we provide a more detailed examination of the electron mass generation.

The cubic coupling $\bar{H_{3}}H_{-}S_{1}^{*}$ results in three physical charged scalars participating in the one-loop diagram ( Fig. 3 ) responsible for electron mass generation. We label these charged scalars as $\eta_{i}^{+}$ for $i=1,2,3$. These charged scalars are related to $\tilde{\eta}=\{H_{3}^{+},H_{-}^{+},C_{4}^{+}\}$ via an orthogonal transformation, such that $\tilde{\eta}_{a}=G_{ai}\eta_{i}^{+}$. In this context, $G$ represents the diagonalization matrix for the charged scalar mass-squared matrix

$$\begin{pmatrix}M_{3}^{2}&\mu_{-,3,s}\frac{v_{1}}{\sqrt{2}}&-\frac{\lambda_{e}v_{1}v_{0}}{2}\\ \mu_{-,3,s}\frac{v_{1}}{\sqrt{2}}&M_{-}^{2}&0\\ -\frac{\lambda_{e}v_{1}v_{0}}{2}&0&M_{4}^{2}\end{pmatrix}$$

(67)

in the basis of $\tilde{\eta}$.

It is observed that $H_{3}^{+}$ and $H_{-}^{+}$ exhibit sizable mixing, whereas the mixing between $H_{3}^{+}$ and $C_{4}^{+}$ is much smaller. And the interaction between electron and the vector fermion $N$ can be expressed as ${\cal L}\supset\bar{N}(\alpha_{i}\hat{L}+\beta_{i}\hat{R})e\,\eta_{i}^{+}+H.C.$, where $\alpha_{i}=y^{e}_{3}G_{1i}$ and $\beta_{i}=y^{e}_{C}G_{3i}$. The one-loop contribution to the electron mass is finite and is expressed as

$$\delta m_{e}={M_{N}\over 16\pi^{2}}\sum_{i=1}^{3}\alpha_{i}\beta_{i}^{*}{\rho_{i}\ln\rho_{i}\over 1-\rho_{i}}\,,$$

(68)

where $\rho_{i}=(M_{\eta_{i}}/M_{N})^{2}$.

Furthermore, the one-loop BSM contribution to the anomalous magnetic moment can be calculated as follows [37]:

$$\Delta a^{BSM}_{e}=\frac{1}{16\pi^{2}}\frac{m_{e}}{M_{N}}\sum_{i=1}^{3}\mathcal{R}(\alpha_{i}\beta_{i}^{*})I_{2}(\rho_{i})\,,$$

(69)

where

$$I_{2}(x)=\frac{1-x^{2}+2x\ln x}{(x-1)^{3}}\,.$$

(70)

Considering the radiative generation of the electron mass, $m_{e}=\delta m_{e}\simeq 0.511\text{ MeV}$, which is a physical and real number, the phase of $y^{e}_{3}(y^{e}_{C})^{*}$ must be removed through field redefinition. As a robust prediction, this model asserts $\Delta a^{BSM}_{e}<0$, although the one-loop correction, $|\Delta a^{BSM}_{e}|\simeq\frac{m_{e}^{2}}{\Lambda^{2}}\sim\mathcal{O}(10^{-21})$, is too small to explain the observed value[38]. Additionally, the one-loop electric dipole moment of the electron, $\propto\Im[y^{e}_{3}(y^{e}_{C})^{*}]=0$, vanishes.

In this model, the new BSM fields to which the electron couples are $N$, $C_{4}$, $H_{3}$, and $Z_{F}$. Moreover, there are only four distinguished topologies of two-loop diagrams [39, 40]. An exhaustive but straightforward examination of all two-loop Feynman diagrams confirms $d_{e}=0$ up to the two-loop level, which automatically mitigates the CP problem, rendering the BSM contribution to $d_{e}$ negligible.

One can proceed to calculate the effective Yukawa coupling of the electron to the SM Higgs, $h^{0}_{SM}$. This calculation incorporates additional parameters in the scalar sector and may lead to a detectable deviation from the SM prediction, $y_{e}^{SM}=m_{e}/v_{0}$. Here, we denote the cubic coupling as $\mu_{ij}\eta_{i}^{*}\eta_{j}h^{0}_{SM}$, involving a dimensionful parameter $\mu_{ij}$.

The effective electron Yukawa coupling can then be determined as

$$y_{e}^{eff}=\sum_{ij}{\alpha_{i}^{*}\beta_{j}\over 16\pi^{2}}\frac{\mu_{ij}}{M_{N}}I_{3}(\rho_{i},\rho_{j},\rho_{q})$$

(71)

where $\rho_{q}=q^{2}/M_{N}^{2}$, with $q$ being the 4-momentum of the SM Higgs, and the electron mass is ignored. The integral function is defined as

$$I_{3}(a,b,c)=\int^{1}_{0}dx\int^{1-x}_{0}\!dy\,{1\over 1-x-y+xa+yb-xyc}\,.$$

(72)

Assuming that $q^{2}\ll M_{N}^{2}$, the integral simplifies to

$$I_{3}(a,b,0)={1\over a-b}\left({a\ln a\over a-1}-{b\ln b\over b-1}\right)\,,$$

(73)

and it approaches the limit ${a-1-\ln a\over(a-1)^{2}}$ when $a=b$.

Regarding the cubic couplings $\mu_{ij}$, the strengths of the quartic couplings $|H_{3}|^{2}|H|^{2}$, $|H_{-}|^{2}|H|^{2}$, and $|C_{4}|^{2}|H|^{2}$ also contribute. Specifically, the cubic couplings are given by

$$\mu_{ij}=\frac{\lambda_{e}v_{1}}{2}G_{1i}G_{3j}+\lambda_{3}v_{0}G_{1i}G_{1j}+\lambda_{-}v_{0}G_{2i}G_{2j}+\lambda_{4}v_{0}G_{3i}G_{3j}\,,$$

(74)

where $\lambda_{3}$, $\lambda_{-}$, and $\lambda_{4}$ are the corresponding quartic coupling constants. However, their significance is highly suppressed by the factor $v_{0}/v_{1}$.

Unlike the cases studied in [41, 35, 36], there are no tree-level contributions to the electron mass, and the mixing between $H_{3}^{+}$ and $C_{4}^{+}$, $\sim(v_{1}v_{0}/\Lambda^{2})\simeq 10^{-6}$, is exceedingly small. Consequently, the resulting effective electron Yukawa coupling is expected to match the SM value. To verify this, we conducted a numerical scan to explore the effective electron Yukawa coupling to the SM Higgs. By requiring the correct electron mass, we allowed the parameter $M_{N}$ to vary between $1\,\mathrm{PeV}$ and $50\,\mathrm{PeV}$, while $\lambda_{3}$, $\lambda_{-}$, and $\lambda_{4}$ were varied between $0.01$ and $\sqrt{4\pi}$. As expected, the deviation from the SM prediction is undetectable.