Democratic parameterization and analysis for 331 model as a subgroup of $SU(6)$

Although the Standard Model (SM) is effective in accurately describing all fundamental forces excluding the gravity, it suffers from large mass spectra and fermionic hierarchies, small quark mixing angles, and the existence of three fermion generations, violation of CP, etc. A number of extensions to SM have been considered to address some of these issues. The so-called 331 model is one of the simplest extensions that change the electroweak gauge group of SM from $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ to $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$. Initially, these models were presented as a natural explanation for the number of fermionic families observed in nature.

Many research have focused on the 331 based model, which was inspired by the need to solve issues in numerous phenomenological applications. For example, papers on 331 model include, but not limited to, applications in neutrino mass generation Boucenna:2014ela ; Tully:2000kk , flavour physics Duy:2022qhy ; Addazi:2022frt ; Buras:2013dea ; Buras:2012dp , and another phenomenological challenges Singer:1980sw ; Pisano:1992bxx ; Frampton:1992wt ; Reig:2016tuk ; Long:1995ctv ; CarcamoHernandez:2005ka ; Liu:1993gy ; Profumo:2013sca ; CarcamoHernandez:2013krw ; Fonseca:2016tbn . Beyond that, taking into account by now well recognized W boson mass anomaly, reported earlier this year by the Collider Detector at Fermilab (CDF) Collaboration obtained at Tevatron particle accelerator CDF:2022hxs , probable interconnection between W mass anomaly and the super-symmetric version of the 331 model has been studied Rodriguez:2022wix . For more up to date articles on 331 models check CarcamoHernandez:2013zrj ; CarcamoHernandez:2014wdl ; CarcamoHernandez:2017cwi ; Barreto:2017xix . From another side, models based on 331 gauge group may be interpreted as a forerunner to grand unification models at high energy scales Deppisch:2016jzl ; Kownacki:2017uyq ; Kownacki:2018lkj . At last, 3311 model, an extended alternative of the 331 model, has been studied in the context with neutrino mass generation mechanism and dark matter candidates Alves:2016fqe ; Dong:2017zxo ; Kang:2019sab ; Leite:2019grf .

Various variations of the 331 type have been studied in detail so far. This model can be made anomaly free in a variety of ways. Model 331 can be made anomaly free within the family like SM. Alternatively, other variants can use all three families and be anomaly free. The second approach is very attractive because it naturally explains SM’s family number of three.

In a recent article submitted by authors of the present paper Ciftci:2022lrc , Democratic Mass Matrix (DMM) approach has been applied to $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ extension of the SM inspired by $E_{6}$ symmetry. The model, named as Variant-A, is anomaly free per generation of quarks and leptons Sanchez:2001ua . In the same work, another anomaly free model in quark/lepton generation, named as Variant-B, is given as $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ extension of the SM and is inspired by both $SU(6)$ Hartanto:2005jr and $SU(6)\otimes U(1)_{X}$ Martinez:2001mu2 symmetries. While Variant-A has additional (isosinglet) quarks in down sector, Variant-B has additional (isosinglet) quarks in up sector. Details of the latter variant are given in Ref. Ponce:2001jn .

Introductory literature on 331 models and DMM approach can be reached in our mentioned paper Ciftci:2022lrc . Yukawa coupling constants of the weak interaction Lagrangian are assumed by DMM to be about the same. Fermions acquire various masses as a result of the small deviations from the full democratic mass matrices through calculation of mass eigenvalues. The democratic parameterization of Variant-A was very successful to fit into the recent experimental data Fritzsch:2021ipb on quark masses and CKM mixing matrix at the scale of mass of Z boson Ciftci:2022lrc . In this study, we want to confirm that the same parameterization works for the Variant-B of 331 model as well. Since a summary of motivation and short history of the DMM approach and 331 Model is given in our earlier paper, we will not go over it again.

The structure of the paper is as follows: The quark content and new gauge bosons, neutral and charged currents, DMM parameterization of the variation B of the 331 model are given in Section 2. Section 3 contains details and definition of the new parameterization for the B variant. Numerical analysis and generated correlation graphs are given in section 4. Analysis results, more precisely the input parameters and obtained observable variables for the three most important and relevant benchmark points are presented in section 5. Future prospects and features of the collected results are discussed in Section 6. Conclusion is given in section 7.

The model with $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ electroweak gauge group is one of the minimal extensions of SM. It is possible to envisage various sub-models of this model with no exotic electrically charged particles Ponce:2001jn . Triangle anomalies can be eliminated throughout each generation. In fact, one of these models (Variant-A) was taken into consideration by the authors of this work in an earlier paper. Since this study’s focus is on the prospect of democratic parameterization of another model (Variant-B) with anomalies canceling in each generation independently, a brief overview of the model’s quark sector, charged currents, and neutral currents is provided in this sub-section.

Every quark mass matrix has a little variation, symbolized by parameters labeled as $\beta$ and $\gamma$, which breaks the democratic pattern. The shape of the deviation for down, up, and heavy up BSM quarks comprises identical structure. Nevertheless, separate parameter sets are used to parameterize variances. The following are the quark mass matrices of down, up, and heavy up BSM isosinglet sectors

In addition, quarks of the up sector and BSM isosinglet up quarks mix with each other, and the mixing is parameterized by $\varepsilon$ parameter according to Eq. (2.3):

$\mathcal{M}^{0}_{uU}$ on the $SU(3)_{L}\otimes U(1)_{X}$ basis, $6\times 6$ mass matrix diagonalization, generates masses of up SM and Beyond Standard Model (BSM) isosinglet quarks on the mass basis. This mass matrix can be diagonalized with the help of a $6\times 6$ unitary matrix $U_{uU}$. While down sector quark masses are obtained by diagonalizing $\mathcal{M}^{0}_{d}$ mass matrix with a $3\times 3$ unitary matrix $U_{d}$. In a similar manner, $3\times 3$ mixing matrices, $U_{u}$ and $U_{U}$, for up type SM and heavy BSM quarks, respectively, are defined as unitary matrices that diagonalize u and U blocks of the $\mathcal{M}^{0}_{uU}$ given in Eq. (13). For simplicity, CP violating phases are considered to be zero from now on. As a result, diagonalizing matrices are real orthogonal matrices.

The mixing matrices $V^{W}_{CKM}$, $V^{K^{\pm}}$ and $V^{K^{0}}$ correspond to W boson of SM, whereas $K^{\pm}$ and $K^{0}$ are heavy BSM gauge bosons, respectively. These mixing matrices are defined through a combinations of $3\times 3$ diagonalizing matrices $U_{u}$, $U_{d}$, and $U_{U}$, mentioned above, and are given by

These matrices can be parameterized with three mixing angles and one phase angle:

here $s_{ij}\equiv\sin\left(\theta_{ij}\right)$, $c_{ij}\equiv\cos\left(\theta_{ij}\right)$, $\theta_{ij}$ are the mixing angles, and $\delta$ is CP violating phase(not taken into account in the present work).

The analysis performed over the model parameterization can be divided into three stages: a systematic scan over all, seven in total (for details see Tab. 1), input parameters of the model, next a more fine grained scan near the points with minimal deviation from the experimental data is performed, then the obtained results were used as a input data for the neural network (NN) training, and further, for obtaining a complete scan over input parameter range. Following the numerical scans, the correlation analyses between different input parameters, distinctive input parameters and predicted observable variables, as well as between various output observable variables was performed. The purpose of studying these correlations is to increase the predictive power of the model and assist in probing the model in the current and future phenomenological experiments. Included below are some of the most important and relevant correlations between input parameters and/or observable variables. The task of the present section is to study the origin behind these correlations.

Results shown below were obtained with the following values for $a$ and $\eta$ (defined in Sec. 2.3) parameters

The strongest correlation patterns between distinct input parameters are shown in Fig. 1. As can be seen from Fig. 1(a), there is direct correlation between $\beta_{d}$ and $\gamma_{d}$ input parameters. Fig. 1(b) demonstrates the correlation between $\gamma_{u}$ and $\gamma_{d}$, which exhibits an inverse correlation contrary to the $\beta_{d}$ vs $\gamma_{d}$ case. Both of these behaviours a drastically different from analogous correlation for the Variant A of the 331 model Ciftci:2022lrc . Other combinations of input parameters exhibit no apparent correlation.

The plots in the Fig. 4 demonstrate important dependence of some observable variables on model input parameters. For instance, plot in Fig. 2(a) shows the direct but weak dependence of $m_{d}$, lightest eigenvalue of the down quark sector, on the input parameter $\beta_{d}$, compared to the cases with its heavier counterparts of the sector. Strong direct correlations are observed between $m_{s}$ and $\beta_{d}$, as well as between $m_{b}$ and $\beta_{d}$ (Fig. 2(b), 2(c)). Furthermore, from Fig. 2(d) one can see that there is a similar linear dependence of $\sin\left(\theta_{23}^{\text{\tiny{CKM}}}\right)$ on $\beta_{d}$. An analogous correlation can be seen between $\gamma_{d}-m_{d}$, $\gamma_{d}-m_{s}$, $\gamma_{d}-m_{b}$ and $\gamma_{d}-\sin\left(\theta_{23}^{\text{\tiny{CKM}}}\right)$ (Fig. 3(a),  3(b),  3(c) and Fig. 3(d)), which exhibit proportional direct-linear behaviour. The direct proportionality between $m_{d}$ and $\gamma_{d}$ can be seen from Eq. (12h), for which the lightest eigenvalue ($m_{d}$) approaches to zero as $\gamma_{d}$ goes to zero. Since $m_{s}$ and $m_{b}$ are most sensitive to the $\beta_{d}$ their $\beta_{d}$ plots are much thinner compared to their plots vs $\gamma_{d}$. Situation with $m_{d}$ is reversed because $\gamma_{d}$ has a leading effect on it. Similar to the situation in the down sector Fig. 2(b), $\beta_{u}$ has a strongest influence on the $m_{c}$, Fig. 4(a), with a direct-linear behaviour. Furthermore, an inverse proportionality between $m_{d}$ and $\gamma_{u}$, which is depicted in Fig. 4(b), is originated from indirect relation between up and down sectors through the CKM mixing angles.

From the above analysis it can be concluded that $\beta_{d}$ and $\gamma_{d}$ have noticeable influence on all the down sector mass eigenvalues and $\sin\left(\theta_{23}^{\text{\tiny{CKM}}}\right)$, whereas $\beta_{u}$ affects up sector quark masses. The correlation between $m_{t}$ and $\beta_{u}$ is absent due to the mixing of SM up quark sector with BSM heavy quarks. $\gamma_{u}$ has the strongest effect on $m_{u}$. Fig. 4(c) demonstrates two minima of $m_{u}$ with respect to $\gamma_{u}$. Similarly, two minima are observed in other input parameters vs $m_{u}$ plots. $m_{u}$ dependence on $\gamma_{u}$ is not direct nor linear, unlike the situation with $\gamma_{d}$ and $m_{d}$ (Fig. 3(a)), this is caused by the affect of $\varepsilon$ and mixing of $m_{u}$ with heavy $m_{U}$ state. Figs. 3(a) and  4(c) contain similar patterns in a sense that patterns consist of two disconnected minimal regions.

Among remarkable correlation patterns of the mixing angles of the CKM matrix are expressed in the $\sin\left(\theta_{23}^{\text{\tiny{CKM}}}\right)$ mixing angle, proportional linearly and totally constrained by input parameters $\gamma_{d}$ and $\beta_{d}$ (Fig. 3(d) and Fig. 2(d)). Input variables demonstrate more complicated effect on the other CKM mixing angles and therefore will be skipped in the further discussion.

SM fermion masses and CKM mixing angles dependence on the mixing parameter($\varepsilon$) between SM up quarks and heavy BSM counterparts is depicted in Fig. 5. $\varepsilon$ has a unique effect on the mass eigenvalues. For the up sector, the lightest mass eigenvalue, $m_{u}$, dominantly depends on the $\gamma_{u}$, whereas dependence of $m_{c}$ is lead by $\beta_{u}$, and top quark mass, $m_{t}$, is linearly dependent on $\varepsilon$, Figs. 5(a), in the neighborhood of $\varepsilon\rightarrow 1$ and approaches zero in its limit. This can be easily seen when $\beta_{u},\gamma_{u}\rightarrow 0$ and taking limit $\varepsilon\rightarrow 1$

Since the $\varepsilon$ parameter only controls the mixing in the up sector, the down sector practically is independent of it, Figs. 5(b). The only indirect effect can be observed via the CKM mixing. Finally, for the CKM mixing angle dependence on $\varepsilon$ one can observe indirect relation via the mixing matrix in the up sector. This can be explained in the following way, since top quark mass, $m_{t}$, strongly depends on $\varepsilon$, changing which will alter the top quark mass, it will then also shift the values of mixing angles between first two and third family in the up sector, which in return will reveal itself in the CKM mixing matrix, Fig. 5(c).

Plots in the Figs. 6(a), 6(b) and  6(c) demonstrate a direct-linear correlation between all three down sector quark masses. This is an immediate consequence of the fact that all three strongly depend on the $\beta_{d}$ input parameter, Figs. 2(a), 2(b), and  2(c). A second local minimum of the best fit of the model in Fig. 6(d) is the same minimum that appeared in Figs. 3(a) and 4(c).

Figs. 7(a), 7(b) and  7(c) show the correlations between down quark sector masses and CKM mixing angle, $\sin\left(\theta_{23}^{\text{\tiny{CKM}}}\right)$. This can be seen immediately from the direct linear dependence of $\sin\left(\theta_{23}^{\text{\tiny{CKM}}}\right)$ on $\beta_{d}$ and $\gamma_{d}$, Fig. 2(d) and Fig. 3(d), respectively. Furthermore, $m_{d}$, $m_{s}$, and $m_{b}$ all depend linearly on $\beta_{d}$ with various level of strength, Figs. 2(a), 2(b), and 2(c). Finally, in Fig. 7(d) one can observe a "star"like pattern similar to the one given in the $\varepsilon$ vs $\sin\left(\theta_{12}^{\text{\tiny{CKM}}}\right)$ plot of Fig. 5(c). This similarity arises from the linear behaviour of $m_{t}$ on $\varepsilon$ in the limit $\varepsilon\rightarrow 1$, eq.(17).

Following the phases assessment in the text succeeding the Eq. (13), in the situations when output observable variables were generated with a negative sign, the later was omitted.

The results of the model predictions are given and elaborated on in the present section. This interpretation of 331 model, inspired by $SU(6)\otimes U(1)$, anticipates SM up and down quark masses, along with, CKM mixing angles for total of seven input parameters. Down, up and up type BSM isosinglet quark sectors are regulated by two parameters each. In addition, light and massive BSM up quarks are mixed with an additional parameter denoted as $\varepsilon$. The Tab. 1 provides collective list of the input parameters for the three most appropriate and significant benchmark points. The first benchmark point (BP1) is described as a point with the smallest $\chi^{2}$ of roughly $0.668$, which has maximum deviation from experimental results of $0.611\sigma$  Eq. (18). The second benchmark point (BP2), contrasted with the first, is defined as the position in a parameter space scan with the lowest overall collection of deviations for all nine observable variables at present with a maximum deviation of $\sim 0.487\sigma$. Finally, we give the average of all data points obtained by $\forall\sigma_{\text{max}}\leq 1$ as the third benchmark point (BP3), designated as BP3_⟨⟩ in Tab . 1, while the spread (error) of all points contributing to $\forall\sigma_{\text{max}}\leq 1$ is expressed as Spread. The deviation $\sigma$ is described as follows

here $x$ indicates any of the observable variables from Tab. 2, exp. stands for the experimentally obtained value, th corresponds to the simulated observable value from the run of the parameter space scan, and lastly, err. means the error for the experimentally obtained value.

Parameter scanning is very sensitive to the precision of input parameter values, so their values are given in Tab. 1 with up to twenty decimal places. The best result for $\chi^{2}$ for the sum of seven input parameters is given in columns 4 and 5 of the table. 2 with a $\chi^{2}\approx 0.668$. As is observed, $m_{u}$ contributes the most to the $\chi^{2}$, but the third generation quark masses of up and down sectors generate a significantly lower imprecision to the $\chi^{2}$. Then, as a result of finding the smallest combination of the deviations from the experimental values (2nd and 3rd columns of Tab. 2), the best obtained point is given in the 6th and 7th columns from Tab. 2 with $\chi^{2}\approx 1.257$ and $\sigma_{\text{max}}\approx 0.487$. Finally, we collect all points with maximum deviations ($\sigma_{\text{max}}\leq 1$) to generate mean and spread values for the set of the observable variables, given in the 8’th and 9’th column of Tab. 2 with $\chi^{2}\approx 1.819$. These numbers represent the location and size of the region, with deviations from the experimental values smaller than one (green area in Fig. 8).

The masses and mixing angles in Tab. 2 were defined as eigenvalues of mass matrices in Eqs. (12h), (13), and as in Eq. (15) for $V^{W}_{CKM}$(Eq. (14d)), $V^{K^{\pm}}$(Eq. (14h)), $V^{K^{0}}$(Eq. (14l)), respectively.

Figure 8 summarizes all the data points collected during input parameter space scan according to two criteria: horizontal axis corresponds to $\sigma_{\text{max}}$ which represents the maximum deviation of each data point with respect to the experimental value obtained up to date, whereas the vertical axis shows the corresponding $\chi^{2}$ values for each data point obtained. The plot in Fig. 8 is divided vertically into three horizontal regions according to the value of $\sigma_{\text{max}}$: $0-1$, $1-2$, $2-3$; vertical region is separated into three categories as well, according to the values of $\left\langle\sigma\right\rangle/\sigma_{\text{max}}$: $0-0.33$, $0.33-0.5$, $0.5-1.0$. The subdivision according to the last category represents the spread of all errors that contribute the total $\chi^{2}$. The solid curves on the plot stand for upper and lower theoretical limits for this plot given by $\chi^{2}=9\sigma_{\text{max}}^{2}$ and $\chi^{2}=\sigma_{\text{max}}^{2}$, respectively.

The plots shown in the previous section, Figs. 4 and 7, can be used to identify and determine the reasons of varying levels of correlation between parameters and observable variables. As expected, the $\gamma$ and $\beta$ parameters have an effect on the mass values of quarks in the up and down sectors. For example, $\gamma_{u}$ is expected to be strongly correlated with $m_{u}$, and $m_{c}$ and $m_{t}$ are expected to be weakly correlated. However, because $\beta u$ is about 70 times bigger than $\gamma_{u}$, the strong correlation of $\gamma_{u}$ with $m_{u}$ is blurred into medium level due to $\beta_{u}$ interference. $\beta_{u}$, as predicted, has a strong correlation with $m_{c}$ and $m_{t}$. It has a little effect on $m_{u}$ due to the relative size of $\beta_{u}$ in comparison to $\gamma_{u}$. The presence of BSM heavy isosinglet quarks (henceforth the BSM effect), is another factor in determining the masses of the SM up sector quarks. This effect is governed in the model by the input parameter $\varepsilon$, which is the dominant influence parameter on the $m_{t}$ mass and in the limit $\varepsilon$ approaches one $m_{t}$ vanishes.

On the other hand, the situation differs drastically for the down sector of the SM. $\gamma_{d}$ and $m_{d}$ are correlated on a medium level, much like the up sector. The leading effect of $\beta_{d}$ results in a subdominant correlation between $\gamma_{d}$ and the heavier down quark mass eigenvalues ($m_{s}$ and $m_{b}$). Correlation of $\beta_{d}$ with $m_{d}$, $m_{s}$, and $m_{b}$ is enhanced proportionally to the mass of the down sector quark, because the effect of $\gamma_{d}$ becomes less apparent with larger mass of the quark. Since there is no BSM effect in strong contrast with the up SM sector, lighter mass eigenvalues’, e.g. $m_{d}$, dependence is lead by $\gamma_{d}$, whereas larger mass eigenvalues’, e.g. $m_{b}$, dependence is dominated by $\beta_{d}$.

Regarding the CKM mixing angles, only the $\sin\left(\theta_{23}^{\text{\tiny{CKM}}}\right)$ has a correlation pattern with $\gamma_{d}$, $\beta_{d}$ input parameters and down sector SM quark masses. The correlation of other CKM mixing angles with input parameters or SM quark masses is much weaker or not observed at all.

As previously stated, CP violating phases are not taken into account in the present paper and left for consideration elsewhere. As a result, the elements of the mass matrices are selected to be real numbers. Therefore, some of the eigenvalues of mass matrices and some elements of the CKM matrix are obtained as negative. By adding phase multipliers to the democratic mass matrix elements, these negative signs can be removed and correct CP violating phases can be obtained. These multipliers are expected to help determine the values of $\chi^{2}$ and $\sigma_{\text{max}}$ as close to zero as possible. The effect of the phases on the quark masses and CKM mixing angles will be further investigated in the future.

The utilization of the DMM technique to the quark sector of the $SU(6)$ symmetry motivated 331 model Variant-B is the subject of current work. Model stands out as one of the simplest extensions of SM. Using a total of ten parameters, the quark masses and mixing angles can be obtained within one standard deviation of the experimental values. A set of three parameters ($a,\gamma,\beta$) primarily control each quark sector (up, down and isosinglet up). In addition, one of the three parameters controls how SM and BSM isosinglet up type quarks are mixed. Therefore, all masses and mixing angles of SM and BSM isosinglet quarks are successfully predicted. There are a total of eighteen observable variables, nine of which are SM variables.

The best fit benchmark points are obtained by performing detailed analysis. It is found that, the best fit point is the point with the lowest $\chi^{2}=0.668$ and the maximum standard deviation $0.611$ from the experimental value, which is corresponding to $m_{u}$. The other critical benchmark point is the one which has the lowest achievable error of standard deviation from the experimental data, with a $\chi^{2}=1.257$ value and the lowest maximum deviation of $0.487$. Besides producing the point with mean value for all created data set with the $\sigma_{\text{max}}\leq 1$ condition, the plot summarizing all the data set in $\sigma_{\text{max}}$ vs $\chi^{2}$ graph is also generated.

In our previous paper, the democratic parameterization Variant-A of 331 model has succeeded in guiding us to the SM quark masses and hierarchy between them in accordance with the recent experimental data. Among the goals of this study, is to confirm that the parameterization at hand is valid for the Variant-B as well. The current work proves that SM quark masses and hierarchy among them are capable of being produced successfully via the democratic parameterization. Additionally, CKM mixing angles are also obtained within appropriate experimental limits. This leads us to a conclusion that further studies on altered parameter schemes based on fundamental democratic pattern are well motivated. Future research should examine UV models of flavor symmetry that naturally lead to democratic based scheme of quark mass sector. Conclusion drawn from the foregoing is that this may also provide a solution to the hierarchy problem.