Investigation of the Higgs boson anomalous FCNC interactions in the simple 3-3-1 model

Let us consider a non-zero rate for a lepton flavor-violating decay mode of the Higg decay. This phenomenology is directly related to the leptonic part of Eq. (20). In the physical basis for the scalar, this part can be rewritten as follows:

	$\displaystyle\mathcal{L}_{Y}\supset$		$\displaystyle-\bar{e}_{aR}\left(c_{\zeta}\frac{1}{u}\left(\mathcal{M}_{e}\right)_{ab}-s_{\zeta}\frac{h^{\prime e}_{ab}}{\sqrt{2}}\frac{uw}{\Lambda^{2}}\right)e_{bL}h-\bar{e}_{aR}\left(s_{\zeta}\frac{1}{u}\left(\mathcal{M}_{e}\right)_{ab}+c_{\zeta}\frac{h^{\prime e}_{ab}}{\sqrt{2}}\frac{uw}{\Lambda^{2}}\right)e_{bL}H$		(27)
			$\displaystyle-\bar{(e_{aL})^{c}}\left(c_{\theta}h_{ab}^{e}+s_{\theta}h^{\prime e}_{ab}\frac{uw}{2\Lambda^{2}}\right)\nu_{bL}H^{+}+\bar{(\nu_{aL})^{c}}\left(c_{\theta}h_{ab}^{e}\right)e_{bL}H^{+}$
			$\displaystyle+\frac{s^{\nu{\dagger}}_{ab}}{\Lambda}\frac{u}{\sqrt{2}}c_{\theta}\left(\bar{\nu_{aL}}e_{bR}+\bar{(e_{aR})^{c}}(\nu_{bL})^{c}\right)H^{+}+h.c.,$

where $\left(\mathcal{M}_{e}\right)_{ab}=\sqrt{2}u\left(h_{ab}^{e}+\frac{h^{\prime e}_{ab}w^{2}}{4\Lambda^{2}}\right)$ is a mixing mass of charged leptons. We denote $e^{\prime}_{L,R}=\left(e,\mu,\tau\right)_{L,R}=(U^{e}_{L,R})^{-1}\left(e_{1},e_{2},e_{3}\right)_{L,R}$, $\nu_{L}^{\prime}=\left(\nu_{e},\nu_{\mu},\nu_{\tau}\right)_{L}=(V^{\nu}_{L})^{-1}\left(\nu_{1},\nu_{2},\nu_{3}\right)_{L}$, the Lagrangian given in (27) can be rewritten as

	$\displaystyle\mathcal{L}_{Y}\supset$		$\displaystyle\bar{e}^{\prime}_{R}g_{h}^{ee}e^{\prime}_{L}h+\bar{e}^{\prime}_{R}g_{H}^{ee}e^{\prime}_{L}H$		(28)
			$\displaystyle+\left\{\bar{(e^{\prime}_{L})^{c}}g_{L}^{e\nu}\nu^{\prime}_{L}+\bar{(\nu^{\prime}_{L})^{c}}g_{L}^{\nu e}e^{\prime}_{L}+\bar{\nu^{\prime}}_{L}g_{R}^{\nu e}e^{\prime}_{R}+\bar{(e^{\prime}_{R})^{c}}g_{R}^{e\nu}(\nu^{\prime}_{L})^{c}\right\}H^{+}+h.c.,$

where $g_{h}^{ee}=U_{R}^{e{\dagger}}\left(c_{\zeta}\frac{1}{u}\mathcal{M}_{e}-s_{\zeta}\frac{uw}{\sqrt{2}\Lambda^{2}}h^{\prime e}\right)U^{e}_{L}$, $g_{H}^{ee}=U_{R}^{e{\dagger}}\left(s_{\zeta}\frac{1}{u}\mathcal{M}_{e}+c_{\zeta}\frac{uw}{\sqrt{2}\Lambda^{2}}h^{\prime e}\right)U^{e}_{L}$, $g_{L}^{e\nu}=(U_{L}^{e})^{T}\left(c_{\theta}h^{e}+s_{\theta}\frac{uw}{2\Lambda^{2}}h^{e\prime}\right)U_{L}^{\nu}$, $g_{L}^{\nu e}=(U_{L}^{\nu})^{T}c_{\theta}h^{e}U^{e}_{L}$, $g_{R}^{\nu e}=U_{L}^{\nu{\dagger}}c_{\theta}\frac{u}{\sqrt{2}\Lambda}s^{\nu}U^{e}_{R}$, $g_{R}^{e\nu}=U_{L}^{eT}c_{\theta}\frac{u}{\sqrt{2}\Lambda}s^{\nu}U^{\nu T}_{R}$.

In every parenthesis in the line of Eq. (27), the first term is proportional to the charged lepton masses, whereas the second term in general can contain off-diagonal entries. It is the source of the HLFV processes and leads to the $h\rightarrow e_{i}e_{j}$ decays, with $i\neq j$. The branching for this decay process can be written as follows:

$\displaystyle\text{Br}(h\rightarrow e_{i}e_{j})=\frac{m_{h}}{8\pi\Gamma_{h}}\left(|g_{h}^{e_{i}e_{j}}|^{2}+|g_{h}^{e_{j}e_{i}}|^{2}\right),$

(29)

where $\Gamma_{h}\simeq 4\ \text{MeV}$ is the total Higgs boson $h$ decay width, $g_{h}^{e_{i}e_{j}}$ is the Higgs boson $h$ coupling to the charged leptons that we can be obtained from Eq.(28). This coupling not only depends on the VEVs and the energy scale $\Lambda$ but also on the Higgs couplings $\lambda_{2},\lambda_{3}$.

The numerical result is shown in Fig.1 for fixing $u=246\ \text{GeV},w=\Lambda$. It is easy to see that the branching ratio of the $h\rightarrow\mu\tau$ can reach the experimental $95\%$ C.L. upper bounds on the HLFV branching ratios from the CMS Collaborations and also can be as low as $10^{-8}$. It depends quite strongly on the factor $\frac{\lambda_{3}}{\lambda_{2}}$, $h^{\prime e}$, and on the energy scale $\Lambda$. In the small $\Lambda$ region and for the factor $\frac{\lambda_{3}}{\lambda_{2}}>1$, the branching ratio for $h\rightarrow\mu\tau$ can reach $10^{-3}$. However, in this region, the mixing angle of $\xi$ is large. Thus, the simple 3-3-1 model may face stringent constraints such as the Higgs boson couplings to fermions and gauge bosons. If $\Lambda$ is taken to be a few TeV but below the Landau pole, $\lambda_{1},\lambda_{2}$ are of the same order, the mixing angle $\xi$ is small and the branching ratio for $h\rightarrow\mu\tau$ reaches $10^{-5}$.

We would like to note that the interaction terms which are given in (28) including the lepton flavor-violating and -conserving couplings can affect other LFV processes such as $e_{i}\rightarrow e_{j}\gamma$. Besides this contribution, the charged current interactions also induce LFV processes. In the simple 3-3-1 model, the charged current interactions have the following form:

$\displaystyle-\frac{g}{\sqrt{2}}\left(\bar{\nu}_{aL}\gamma^{\mu}e_{aL}W^{+}_{\mu}+\bar{\nu}_{aL}\gamma^{\mu}e_{aR}^{c}X_{\mu}^{+}+\bar{e}_{aL}\gamma^{\mu}e_{aR}^{c}Y^{--}_{\mu}\right)+h.c..$

(30)

Taking all these ingredients into account, the total contribution to the $\tau\rightarrow\mu\gamma$ decay include:

• •

  the one-loop diagram with singly charged gauge bosons and neutrinos in the loop

• •

  the one-loop diagram with doubly charged gauge bosons and charged leptons in the loop

• •

  the one-loop diagram with charged Higgs bosons and neutrinos in the loop

• •

  the one-loop diagram with neutral Higgs bosons and charged leptons in the loop

• •

  two-loop Barr-Zee diagrams with an internal photon and a third generation quark

• •

  two-loop Barr-Zee diagrams with an internal photon and a gauge boson

The first three types of contributions are the same as those of the 3-3-1 model with a new lepton; see HueLFV . The last three types of contributions come from the source of the HLFV processes that are a new contribution and have been not considered in the previous version of the 3-3-1 model HueLFV . The total effective Lagrangian describing the $e_{i}\rightarrow e_{j}\gamma$ decay process is given as

$\displaystyle em_{\tau}\left\{\bar{e^{\prime}}_{i}\left(D_{R}^{\gamma}\right)_{ij}\sigma^{\alpha\beta}P_{R}e_{j}^{\prime}F_{\alpha\beta}+\bar{e^{\prime}}_{i}\left(D_{L}^{\gamma}\right)_{ij}\sigma^{\alpha\beta}P_{L}e_{j}^{\prime}F_{\alpha\beta}\right\}.$

(31)

It leads to the branching ratio of the $\tau\rightarrow\mu\gamma$ following processes:

$\displaystyle\text{Br}(\tau\rightarrow\mu\gamma)=$

$\displaystyle{}\frac{48\pi^{3}\alpha}{G_{F}^{2}}\left(|D_{L}^{\gamma}|^{2}+|D_{R}^{\gamma}|^{2}\right)Br(\tau\rightarrow\mu\bar{\nu}_{\mu}\nu_{\tau}),$

(32)

where $D_{L,R}^{\gamma}$ comes from the one-loop and two-loop diagrams. Firstly, the one-loop diagram contributions with charged Higgs boson $H^{\pm}$, charged gauge boson $W^{\pm}$ and doubly charged gauge boson $Y^{\pm\pm}$ have a form inspired by the general formula in Lavoura :

	$\displaystyle D_{1R}^{\nu W^{\pm}}=-\frac{eg^{2}m_{\tau}}{32\pi^{2}m_{W}^{2}}\sum_{j=1}^{3}U^{\nu}_{j3}U^{\nu*}_{j2}f\left(\frac{m_{\nu_{j}}^{2}}{m_{W}^{2}}\right),\hskip 14.22636ptD_{1L}^{\nu W^{\pm}}=-\frac{eg^{2}m_{\mu}}{32\pi^{2}m_{W}^{2}}\sum_{j=1}^{3}U^{\nu}_{j3}U^{\nu*}_{j2}f\left(\frac{m_{\nu_{j}}^{2}}{m_{W}^{2}}\right),$
	$\displaystyle D_{1R}^{\nu X^{\pm}}=-\frac{eg^{2}m_{\tau}}{32\pi^{2}m_{X}^{2}}\sum_{j=1}^{3}U^{\nu}_{j3}U^{\nu*}_{j2}f\left(\frac{m_{\nu_{j}}^{2}}{m_{X}^{2}}\right),\hskip 14.22636ptD_{1L}^{\nu X^{\pm}}=-\frac{eg^{2}m_{\mu}}{32\pi^{2}m_{X}^{2}}\sum_{j=1}^{3}U^{\nu}_{j3}U^{\nu*}_{j2}f\left(\frac{m_{\nu_{j}}^{2}}{m_{X}^{2}}\right),$
	$\displaystyle D_{1R}^{eY^{\pm\pm}}=-\frac{eg^{2}m_{\tau}}{32\pi^{2}m_{Y^{\pm\pm}}^{2}}\sum_{j=1}^{3}\left[g\left(\frac{m_{e_{j}}^{2}}{m_{Y^{\pm\pm}}^{2}}\right)-2f\left(\frac{m_{e_{j}}^{2}}{m_{Y^{\pm\pm}}^{2}}\right)\right],$
	$\displaystyle D_{1L}^{eY^{\pm\pm}}=-\frac{eg^{2}m_{\mu}}{32\pi^{2}m_{Y^{\pm\pm}}^{2}}\sum_{j=1}^{3}\left[g\left(\frac{m_{e_{j}}^{2}}{m_{Y^{\pm\pm}}^{2}}\right)-2f\left(\frac{m_{e_{j}}^{2}}{m_{Y^{\pm\pm}}^{2}}\right)\right],$
	$\displaystyle D_{1R}^{\nu H^{\pm}}=-\frac{eg^{2}}{32\pi^{2}m_{H^{\pm}}^{2}m_{W}^{2}}\sum_{j=1}^{3}\left\{g^{\nu\tau*}_{L}g^{\nu\mu}_{L}m_{\tau}h\left(\frac{m_{\nu_{j}}^{2}}{m_{H^{\pm}}^{2}}\right)\right.$
	$\displaystyle\left.\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt+g^{\nu\tau*}_{R}g^{\nu\mu}_{R}m_{\mu}k\left(\frac{m_{\nu_{j}}^{2}}{m_{H^{\pm}}^{2}}\right)+g^{\nu\tau*}_{L}g^{\nu\mu}_{R}m_{H}r\left(\frac{m_{\nu_{j}}^{2}}{m_{H^{\pm}}^{2}}\right)\right\},$
	$\displaystyle D_{1L}^{\nu H^{\pm}}=-\frac{eg^{2}}{32\pi^{2}m_{H^{\pm}}^{2}m_{W}^{2}}\sum_{j=1}^{3}\left\{g^{\nu\tau*}_{R}g^{\nu\mu}_{R}m_{\tau}h\left(\frac{m_{\nu_{j}}^{2}}{m_{H^{\pm}}^{2}}\right)\right.$
	$\displaystyle\left.\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt\hskip 14.22636pt+g^{\nu\tau*}_{L}g^{\nu\mu}_{L}m_{\mu}k\left(\frac{m_{\nu_{j}}^{2}}{m_{H^{\pm}}^{2}}\right)+g^{\nu\tau*}_{R}g^{\nu\mu}_{L}m_{H}r\left(\frac{m_{\nu_{j}}^{2}}{m_{H^{\pm}}^{2}}\right)\right\}.$

with the functions $f,g,h,k$ and $r$ defined by

	$\displaystyle f(x)$	$\displaystyle=$	$\displaystyle\frac{10-43x+78x^{2}-49x^{3}+4x^{4}+18x^{3}\ln{x}}{12(x-1)^{4}},$
	$\displaystyle g(x)$	$\displaystyle=$	$\displaystyle\frac{8-38x+39x^{2}-14x^{3}+5x^{4}-18x^{2}\ln{x}}{12(x-1)^{4}},$
	$\displaystyle h(x)$	$\displaystyle=$	$\displaystyle k(x)=\frac{1-6x+3x^{2}+2x^{3}-6x^{2}\ln{x}}{12(x-1)^{4}},$
	$\displaystyle r(x)$	$\displaystyle=$	$\displaystyle\frac{-1+x^{2}-2x\ln{x}}{2(x-1)^{3}}.$		(34)

The neutral Higgs contribution to $D_{L,R}^{\gamma}$ via the one-loop diagram is

$\displaystyle D_{1L}^{\gamma}=D_{1R}^{\gamma}=\sqrt{2}\sum_{\phi}\frac{g_{\phi}^{\mu\tau}g_{\phi}^{\tau\tau}}{m_{\phi}^{2}}\left(\ln\frac{m_{\phi}^{2}}{m_{\tau}^{2}}-\frac{3}{2}\right),$

(35)

and the two-loop correction to $D^{\gamma}_{LR}$ being given by Sacha

	$\displaystyle D_{2L}^{\gamma}=D^{\gamma}_{2R}=2\sum_{\phi,f}g_{\phi}^{\mu\tau}g_{\phi}^{ff}\frac{N_{c}Q_{f}^{2}\alpha}{\pi}\frac{1}{m_{\tau}m_{f}}f_{\phi}\left(\frac{m_{f}^{2}}{m_{\phi}^{2}}\right)$
	$\displaystyle-\sum_{\phi=h,H}g_{\phi}^{\mu\tau}g_{\phi}^{GG}\frac{\alpha Q_{G}^{2}}{2\pi m_{\tau}m_{G}^{2}}\left\{3f_{\phi}\left(\frac{m_{G}^{2}}{m^{2}_{\phi}}\right)+\frac{23}{4}g\left(\frac{m_{G}^{2}}{m^{2}_{\phi}}\right)+\frac{3}{4}h\left(\frac{m_{G}^{2}}{m_{\phi}^{2}}\right)+m_{\phi}^{2}\frac{f_{\phi}\left(\frac{m_{G}^{2}}{m_{\phi}^{2}}\right)-g\left(\frac{m_{G}^{2}}{m_{\phi}^{2}}\right)}{2m_{G}^{2}}\right\},$

where $\Phi=h,H$, $G=W^{\pm},X^{\pm},Y^{\pm\pm}$, $f=t,b$, and $Q_{G}$ is an electrical charge of the gauge boson $G$. $g_{\phi}^{\mu\tau},g_{\phi}^{ff},g_{\phi}^{GG}$ are the scalar $\phi$ couplings to $\mu$ $\tau$, two fermions, and two gauge bosons $G$, respectively. The expressions for $g_{\phi}^{ff},g_{\phi}^{GG}$ are given in m331 and for $g_{\phi}^{\mu\tau}$ can be obtained from Eq.(28). The loop functions, $f_{\phi}(z),h(z),g(z)$, are given by Sacha

	$\displaystyle f_{h,H}(z)$		$\displaystyle=\frac{z}{2}\int_{0}^{1}dx\frac{\left(1-2x(1-x)\right)}{x(1-x)-z}\ln\frac{x(1-x)}{z},$
	$\displaystyle h(z)$		$\displaystyle=-\frac{z}{2}\int_{0}^{1}\frac{dx}{x(1-x)-z}\left\{1-\frac{z}{x(1-x)-z}\ln\frac{x(1-x)}{z}\right\}$
	$\displaystyle g(z)$		$\displaystyle=\frac{z}{2}\int^{1}_{0}dx\frac{1}{x(1-x)-z}\ln\frac{x(1-x)}{z}.$		(36)

In the limits $z\gg 1$ and $z\ll 1$, the functions $f(z),g(z)$ and $h(z)$ can approximately be written as follow:

	$\displaystyle z\ll 1,\hskip 14.22636ptf(z)=\frac{z}{2}(\ln{z})^{2},\hskip 14.22636ptg(z))=\frac{z}{2}(\ln{z})^{2},\hskip 14.22636pth(z)=z\ln{z},$
	$\displaystyle z\gg 1,\hskip 14.22636ptf(z)=\frac{\ln{z}}{3}+\frac{13}{18},\hskip 14.22636ptg(z)=\frac{\ln{z}}{2}+1,\hskip 14.22636pth(z)=-\frac{\ln{z}}{2}-\frac{1}{2}.$		(37)

For $z\sim\mathcal{O}(1)$, the functions $f,g,h\sim z$ can be accurately calculated. Let us estimate each kind of diagrams contributing to $\tau\rightarrow\mu\gamma$ via numerical studies. We choose the parameters as follows:

	$\displaystyle\hskip 14.22636ptm_{W}=80.385\ \text{GeV},\hskip 14.22636ptm_{e}=0.000511\ \text{GeV},\hskip 14.22636ptm_{\mu}=0.1056\ \text{GeV},\hskip 14.22636ptm_{\tau}=1.176\ \text{GeV}$
	$\displaystyle\sin^{2}(\theta_{12})=0.307,\hskip 14.22636pt\sin^{2}(\theta_{23})=0.51,\hskip 14.22636pt\sin^{2}(\theta_{13})=0.021,\hskip 14.22636pt\alpha=\frac{1}{137},\hskip 14.22636ptu=246\ \text{GeV}$
	$\displaystyle\Delta m_{12}^{2}=m_{\nu_{2}}^{2}-m_{\nu_{1}}^{2}=7.53\times 10^{-5}\ \text{eV}^{2},\hskip 14.22636pt\Delta m_{23}^{2}=m_{\nu_{3}}^{2}-m_{\nu_{2}}^{2}=2.45\times 10^{-3}\ \text{eV}^{2}$
	$\displaystyle\lambda_{2}=\lambda_{1}=0.09,\hskip 14.22636pts^{\nu}\sim 10^{-10}.$		(38)

The results shown in the Fig. 2 suggest that the two-loop diagrams can provide a dominant contribution to $\tau\rightarrow\mu\gamma$. The Br$(\tau\rightarrow\mu\gamma)$ strongly depends on the lepton flavor-violation coupling $\left[(U^{e}_{R})^{\dagger}h^{\prime e}U_{L}^{e}\right]_{\mu\tau}$. If we choose $\left[(U^{e}_{R})^{\dagger}h^{\prime e}U_{L}^{e}\right]_{\mu\tau}=2\frac{\sqrt{m_{\mu}m_{\tau}}}{u}$, the two-loop contribution to $\tau\rightarrow\mu\gamma$ dominates over the one-loop contribution. However, the branch ratio, Br$(\tau\rightarrow\mu\gamma)$, is only consistent with the predictions of the experiment when the new physical scale is above the Landau pole. If $\left[(U^{e}_{R})^{\dagger}h^{\prime e}U_{L}^{e}\right]_{\mu\tau}=5\times 10^{-4}$, the two-loop contribution can be less important and the main contribution comes from one-loop diagrams with the conversed lepton couplings. In this case, we obtain the upper bound on the new physics scale: $\Lambda>2.4\ \text{TeV}$ from the lower bound on Br$(\tau\rightarrow\mu\gamma)$ of the experiment. Comparing the results given in Figs. 2 and 3, we find that the above conclusions change slightly when the factor $\frac{\lambda_{3}}{\lambda_{2}}$ is changed.

The new physics of the 3-3-1 model contributes to the muon’s anomalous magnetic moments $a_{\mu}$ via the flavor conserving couplings was considered by BHHL ; QRmuon . The 3-3-1 model also has FCNC, so it can make its own contribution to the anomalous magnetic moment. First, we investigate only the contribution of the FCNC to $(g-2)_{\mu}$. There exists a one-loop contribution to $(g-2)_{\mu}$ through flavor-violating couplings of the Higgs to $\mu\tau$. According to the work given in Sacha , the one-loop contribution mediated by the neutral Higgs contribution to $\left(g-2\right)_{\mu}$ can be expressed by

	$\displaystyle(\Delta a_{\mu})^{M331}$		$\displaystyle=\sum_{\phi}\left(g_{\phi}^{\tau\mu}\right)^{2}\frac{m_{\mu}m_{\tau}}{8\pi^{2}}\int_{0}^{8}dx\frac{x^{2}}{m_{\phi}^{2}-x(m_{\phi}^{2}-m_{\tau}^{2})}$		(39)
			$\displaystyle\simeq\sum_{\phi}\left(g_{\phi}^{\tau\mu}\right)^{2}\frac{m_{\mu}m_{\tau}}{8\pi^{2}m_{\phi}^{2}}\left(\ln\frac{m_{\phi}^{2}}{m_{\tau}^{2}}-\frac{3}{2}\right).$

We plot in Fig. 4 the muon’s anomalous magnetic moment $\Delta a_{\mu}^{M331}$ as a function of the self-Higgs coupling $\lambda_{2}$, assuming $\Lambda=2000$ GeV, $\left[(U^{e}_{R})^{\dagger}h^{\prime e}U_{L}^{e}\right]_{\mu\tau}=2\frac{\sqrt{m_{\mu}m_{\tau}}}{u},w=\Lambda,u=246$ GeV. This choice leads to the branching ratio $h\rightarrow\tau\mu$ and can be close to the upper limit value of the experiment or as low as $10^{-5}$ but the flavor-changing interactions of neutral Higgs and two leptons contributed negligibly to $\Delta a_{\mu}^{M331}$; see Fig. 4. We would like to emphasize that the new contribution to the muon magnetic moment $(g-2)_{\mu}$ in the context of the simple 3-3-1 model comes from the doubly gauge bosons $Y^{\pm\pm}$, new singly charged vectors $V^{\pm}$, and new singly charged Higgs $H^{\pm}$. The dominant contribution is the doubly charged gauge boson running in the loop QRmuon . The total doubly charged boson contribution is given by

$\displaystyle\Delta a_{\mu}(X^{\pm\pm})\simeq\frac{28}{3}\frac{m_{\mu}^{2}}{u^{2}+w^{2}}.$

(40)

It is easy to check that an energy scale of symmetry breaking $SU(3)_{L}$ around 2 TeV, 1.7 TeV $<w<$2.2 TeV, is favored as regards explaining the discrepancy of the measured value of muon’s anomalous magnetic moment and the one predicted by the standard model muon ,

$\displaystyle(\Delta a_{\mu})_{EXP-SM}=(26.1\pm 8)\times 10^{-10}.$

(41)

As mentioned in m331 , the LHC constraint over the $Z^{\prime}$ mass in the simple 3-3-1 model is $M_{Z}^{\prime}>2.75$ TeV. It can be translated into the lower bound on the VEV, $w$, as follows: $w>2.38$ TeV. Therefore, the parameter space of w, which is favored for explaining $(\Delta a_{\mu})_{EXP-SM}$, is slightly smaller than the lower limit of the LHC (very close to the LHC’s allowed space). In other words, in the parameter space that allows an interpretation to be made of LHC’s experimental results, the value of the muon’s anomalous magnetic moment is predicted, $(\Delta a_{\mu})_{331}<13.8\times 10^{-10}$. The upper limit is very close to the constraint given in Eq. (41).