1 INTRODUCTION

The discrepancies from the Standard Model (SM) predictions are found in the measurements of the muon anomalous magnetic moment and the rare semi-leptonic decays of \(B\) mesons, \(b\to s\ell\ell\). The latest measurements of \(\Delta a_{\mu}=a_{\mu}^{\textrm{exp}}-a_{\mu}^{\textrm{SM}}\) and the lepton universality observable, \(R_{K}\), confirmed that their values are deviated from the SM prediction by \(4.2\sigma\) [1, 2] and \(3.1\sigma\) [3], respectively. These anomalies may imply that there are new particles at the TeV scale which couple to muons.

In [4, 5], we proposed a model which explains the anomalies in \(\Delta a_{\mu}\) and \(b\to s\ell\ell\) simultaneously.Footnote 1 We introduced a complete vector-like (VL) family of quarks and leptons, and a new vector-like \(U(1)^{\prime}\) gauge symmetry. Only the VL family and the \(U(1)^{\prime}\) breaking scalar, \(\Phi\), carry non-zero \(U(1)^{\prime}\) charges, whereas the three generations of chiral families and the Higgs boson are neutral. A \({Z^{\prime}}\) boson associated with the \(U(1)^{\prime}\) gauge symmetry couples to the SM families via the mixing between the chiral and VL families induced by the \(U(1)^{\prime}\) breaking. In this model, \(\Delta a_{\mu}\) is explained by the 1-loop contributions mediated via the VL leptons and the \({Z^{\prime}}\) boson, and \(b\to s\ell\ell\) is explained by the tree-level exchanging of the \({Z^{\prime}}\) boson. We showed that the anomalies can be explained if the VL leptons and \({Z^{\prime}}\) boson are lighter than 1.5 TeV.

We propose a novel possibility to search for the VL leptons and \({Z^{\prime}}\) boson [11]. We consider the pair production of VL leptons decaying to a \({Z^{\prime}}\) boson followed by the \({Z^{\prime}}\) boson decaying to a pair of muons or muon neutrinos. There are six (four) muons in the final state if both (either of) the \({Z^{\prime}}\) bosons decay to muons. We recast the latest ATLAS search [12] for events with four or more charged leptons.

2 EXPLANATION FOR THE MUON ANOMALIES

2.1 Model

Table 1 Quantum numbers of new fermion and scalar fields
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Table 2 Values of \(\chi^{2}\), selected input parameters and observables at the best fit points A, B, C, and D. The degree of freedom in our analysis is \(N_{\textrm{obs}}-N_{\textrm{inp}}=98-65=33\). The last column shows the experimental central values and their uncertainties. The upper limits on the lepton flavor violating decays are \(90\%\) C.L. limits
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We review the model proposed in [4, 5]. The new particles of our model are given in Table 1. The SM particles are neutral under the \(U(1)^{\prime}\) gauge symmetry. For simplicity, we focus on the charged leptons. The first and third generations of the chiral leptons are omitted, although we studied the model with complete generations and discussed lepton flavor violation. The mass of the VL states and Yukawa interactions are given by

$$\mathcal{L}\supset-m_{L}\overline{L}_{R}L_{L}-m_{E}\overline{E}_{R}E_{L}$$
$${}+y_{\mu}\overline{\mu}_{R}\ell_{L}H+{\kappa^{\prime}}\overline{E}_{R}L_{L}H-\kappa\overline{L}_{R}\tilde{H}E_{L}$$
$${}+\lambda_{L}\Phi\overline{L}_{R}\ell_{L}-\lambda_{E}\Phi\overline{\mu}_{R}E_{L}+\textrm{h.c.},$$
(1)

where \(\tilde{H}:=i\sigma_{2}H^{*}=(H_{-}^{*},-H_{0}^{*})\). The \(SU(2)_{L}\) doublets are defined as

$$\ell_{L}=(\nu_{L},\mu_{L}),\quad H=(H_{0},H_{-}),$$
$$L_{L}=(N_{L}^{\prime},E_{L}^{\prime}),\quad\overline{L}_{R}=(-\overline{E}_{R}^{\prime},\overline{N}^{\prime}_{R}),$$
(2)

and the \(SU(2)_{L}\) indices are contracted via \(i\sigma_{2}\). After symmetry breaking by \(v_{H}:=\langle{H_{0}}\rangle\) and \(v_{\Phi}:=\langle{\Phi}\rangle\), the mass matrix for the leptons is given by

$$\overline{\mathbf{e}}_{R}\mathcal{M}_{e}\mathbf{e}_{L}:=\begin{pmatrix}\overline{\mu}_{R}&\overline{E}_{R}&\overline{E}_{R}^{\prime}\end{pmatrix}$$
$${}\times\begin{pmatrix}y_{\mu}v_{H}&0&\lambda_{E}v_{\Phi}\\ 0&{\kappa^{\prime}}v_{H}&m_{E}\\ \lambda_{L}v_{\Phi}&m_{L}&\kappa v_{H}\end{pmatrix}\begin{pmatrix}\mu_{L}\\ E^{\prime}_{L}\\ E_{L}\end{pmatrix}.$$
(3)

The mass basis is defined as

$$\hat{\mathbf{e}}_{L}:=U_{L}^{\dagger}\mathbf{e}_{L},\quad\hat{\mathbf{e}}_{R}:=U_{R}^{\dagger}\mathbf{e}_{R},$$
$$U_{R}^{\dagger}\mathcal{M}_{e}U_{L}=\textrm{diag}\left(m_{\mu},m_{E_{2}},m_{E_{1}}\right),$$
(4)

where \(E_{1}\) and \(E_{2}\) are respectively the singlet-like and doublet-like VL leptons. We define the Dirac fermions as \(\mathbf{e}:=\left(\mu,E_{2},E_{1}\right)\), \(\left[\mathbf{e}\right]_{i}:=\left(\left[\hat{\mathbf{e}}_{L}\right]_{i},\left[\hat{\mathbf{e}}_{R}\right]_{i}\right)\), where \(i=1,2,3\). The mass matrices for the quarks and the Dirac mass matrices for the neutrinos have the same structure as the charged leptons. We assume that the three generations of right-handed neutrinos have Majorana masses at the intermediate scale, so that the tiny neutrino masses are explained by the type-I see-saw mechanism.

The gauge interactions with the \({Z^{\prime}}\) boson in the mass basis are given by

$$\mathcal{L}_{V}=Z^{\prime}_{\mu}\;\overline{\mathbf{e}}\gamma^{\mu}\left({g}^{Z^{\prime}}_{e_{L}}P_{L}+{g}^{Z^{\prime}}_{e_{R}}P_{R}\right)\mathbf{e},$$
$$g^{Z^{\prime}}_{\mathbf{e}_{L}}={g^{\prime}}U_{L}^{\dagger}Q^{\prime}_{e}U_{L},\quad g^{Z^{\prime}}_{\mathbf{e}_{R}}={g^{\prime}}U_{R}^{\dagger}Q^{\prime}_{e}U_{R},$$
(5)

where \(Q_{e}^{\prime}=\textrm{diag}(0,1,1)\) is the coupling matrix in the gauge basis. Note that the chiral family does not couple to the \({Z^{\prime}}\) boson in the gauge basis, and the coupling arises only in mass basis. Here, \(P_{L}\) (\(P_{R}\)) are the chiral projections onto the left- (right-)handed fermions. \({g^{\prime}}\) is the gauge coupling constant for \(U(1)^{\prime}\).

Fig. 1
figure 1

Diagrams contribute to \(\Delta a_{\mu}\) (left) and the \(b\to s\ell\ell\) decay (right).

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2.2 Muon Anomalies

The diagrams that can explain the anomalies are shown in Fig. 1. The \({Z^{\prime}}\) boson and the physical mode \(\chi\) of the \(\Phi\) boson contribute to \(\Delta a_{\mu}\) as [13, 14]

$$\Delta a_{\mu}\sim\frac{m_{\mu}\kappa v_{H}}{64\pi^{2}v_{\Phi}^{2}}s_{2L}s_{2R}\bigg{(}\sqrt{x_{L}x_{E}}\frac{G_{Z}(x_{L})-G_{Z}(x_{E})}{x_{L}-x_{E}}$$
$${}+\frac{1}{2}\sqrt{y_{L}y_{R}}\frac{{y_{L}}G_{{S}}(y_{L})-{y_{R}}G_{{S}}(y_{R})}{y_{L}-y_{R}}\bigg{)},$$
(6)

where \(s_{A}:=\lambda_{X}v_{\Phi}/M_{X}\), with \(M_{X}^{2}:=m_{X}^{2}+\lambda_{X}^{2}v_{\Phi}^{2}\) (\(X=L,E\) for \(A=L,R\)), is the mixing angle between the chiral and VL families. Here, \(x_{L}:= M_{L}^{2}/m_{Z^{\prime}}^{2}\), \(x_{E}:=M_{E}^{2}/m_{Z^{\prime}}^{2}\), \(y_{L}:=M_{L}^{2}/m_{\chi}^{2}\), \(y_{E}:=M_{E}^{2}/m_{\chi}^{2}\) with \(m_{Z^{\prime}}^{2}=2{g^{\prime}}^{2}v_{\Phi}^{2}\) and \(m_{\chi}\) is the mass of \(\chi\). The loop functions are given by

$$G_{Z}(x):=\frac{x^{3}+3x-6x\ln{(x)}-4}{2(1-x)^{3}},$$
$$G_{S}(y):=\frac{y^{2}-4y+2\ln{(y)}+3}{(1-y)^{3}}.$$
(7)

The value of \(\Delta a_{\mu}\) is estimated as

$$\Delta a_{\mu}\sim 2.9\times 10^{-9}\left(\frac{1.0~\textrm{TeV}}{v_{\Phi}}\right)^{2}$$
$${}\times\left(\frac{\kappa}{1.0}\right)\left(\frac{s_{2L}s_{2R}}{1.0}\right)\left(\frac{C_{LR}}{0.1}\right),$$
(8)

where \(C_{LR}\) is inside the parenthesis of Eq. (6).

For the \(b\to s\ell\ell\) anomaly, the Wilson coefficients are given by [15, 16]

$$C_{9}\sim-\frac{\sqrt{2}}{4G_{F}}\frac{4\pi}{\alpha_{e}}\frac{1}{V_{tb}V^{*}_{ts}}\frac{1}{4v_{\Phi}^{2}}(s_{R}^{2}+s_{L}^{2})\epsilon_{Q_{2}}\epsilon_{Q_{3}},$$
(9)
$$C_{10}\sim-\frac{\sqrt{2}}{4G_{F}}\frac{4\pi}{\alpha_{e}}\frac{1}{V_{tb}V^{*}_{ts}}\frac{1}{4v_{\Phi}^{2}}(s_{R}^{2}-s_{L}^{2})\epsilon_{Q_{2}}\epsilon_{Q_{3}},$$
(10)

where the \({Z^{\prime}}\) boson couplings to the SM doublet quarks are parametrized as

$$\left[g^{{Z^{\prime}}}_{d_{L}}\right]_{ij}\sim\left[g^{{Z^{\prime}}}_{u_{L}}\right]_{ij}\sim-{g^{\prime}}\epsilon_{Q_{i}}\epsilon_{Q_{j}}.$$
(11)

Here, \(\epsilon_{Q_{i}}\) (\(i=1,2,3\)) is the similar quantity as \(s_{L,R}\) and originates from the mixing between the SM and VL quarks, but we now consider the couplings with the second and third generation quarks and these are typically small in contrast to that for muons. The value of \(C_{9}\) is estimated as

$$C_{9}\sim-0.62\left(\frac{1.0~\textrm{TeV}}{v_{\Phi}}\right)^{2}$$
$${}\times\left(\frac{s_{L}^{2}+s_{R}^{2}}{1}\right)\left(\frac{\epsilon_{Q_{2}}\epsilon_{Q_{3}}}{-0.002}\right).$$
(12)

2.3 \(\chi^{2}\) Analysis

Fig. 2
figure 2

Processes with four muons or more.

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Fig. 3
figure 3

Limits from the signal regions with 4 leptons for the singlet-like VL lepton. The colors signify upper bounds on the \(\textrm{BR}\left({E}\to{{Z^{\prime}}\mu}\right)\), e.g., \(\textrm{BR}\left({E}\to{{Z^{\prime}}\mu}\right)>0.5\) is excluded at \((m_{E},m_{{Z^{\prime}}})=(750,200)~\textrm{GeV}\) in the right panel.

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We searched for parameter points which can explain the anomalies without unacceptably changing the other observables consistent with the SM. The \(\chi^{2}\) function is defined as

$$\chi^{2}(x):=\sum_{I\in\textrm{obs}}\frac{\left(y_{I}(x)-y_{I}^{0}\right)^{2}}{\sigma_{I}^{2}},$$
(13)

where \(x\) is a parameter space point, \(y_{I}(x)\) is the value of an observable \(I\) whose central value is \(y_{I}^{0}\) and uncertainty is \(\sigma_{I}\). In this model, there are 65 input parameters and we studied 98 observables including fermion masses, CKM matrix, lepton and quark flavor violation observables, and so on. The full list of observables and the values for the \(\chi^{2}\) fitting are shown in [5]. Table 2shows values of \(\chi^{2}\), selected input parameters and observables at the best fit points A, B, C, and D. At these points, both anomalies are explained successfully, while the flavor violation observables, e.g., \(\mu\to e\gamma\) and \(\Delta M_{s}\) are consistent with the current limits. We did a global analysis for the model, and we found that \(\Delta a_{\mu}\) can be explained only if \(m_{Z^{\prime}}<800~\textrm{GeV}\) and \(m_{E}<1.5~\textrm{TeV}\). Therefore, these new particles will be probed by the LHC experiment as discussed in the next section. Although there are no lower bounds on the branching fractions in the lepton flavor violating decays, we found the relations among the decay modes \(\textrm{BR}\left({\mu}\to{e\gamma}\right)\gg\textrm{BR}\left({\mu}\to{eee}\right)\) and \(\textrm{BR}\left({\tau}\to{\mu\gamma}\right)\sim\textrm{BR}\left({\tau}\to{\mu\mu\mu}\right)\gg\textrm{BR}\left({\tau}\to{e\gamma,e\ell\ell}\right)\) (\(\ell=e,\mu\)) which would be confirmed by the future experiments. In the quark sector, most observables are consistent with the SM predictions assuming unitarity of the CKM matrix, although our model can have non-unitarity of the CKM matrix for the SM families.

3 LHC SIGNALS

The VL leptons and \({Z^{\prime}}\) boson are within the reach of the LHC experiment. As shown in Eq. (12), the \({Z^{\prime}}\) boson typically couples to SM quarks only with small couplings. Hence, the strong constraints from the di-muon resonance search [17] for a \({Z^{\prime}}\) boson can be evaded since the production cross section is suppressed even though the branching fraction to muons is sizable. Therefore, the \({Z^{\prime}}\) boson below \(1~\textrm{TeV}\) is, in general, not excluded by the \({Z^{\prime}}\) search.

The processes which can produce more than four muons are shown in Fig. 2. The signal of more than four muons is produced if either of the \({Z^{\prime}}\) boson decays to a pair of muons from charged VL lepton pair production (left), while both have to decay to muons in the other processes (middle and right) involving the VL neutrino.

We recast the limits obtained in [12] searching for signals with more than four leptons. We have generated events using MadGraph5\(\_\)2\(\_\)8\(\_\)2 [18] based on a UFO [19] model file generated with FeynRules\(\_\)2\(\_\)3\(\_\)43 [18, 20]. The events are showered with PYTHIA8 [21] and then run through the fast detector simulator Delphes3.4.2 [21]. We used the default ATLAS card for the detector simulation, but the threshold on \(p_{T}\) for the muon efficiency formula is changed to \(5\) from \(10\) GeV since muons with \(p_{T}>5~\textrm{GeV}\) are counted as signal muons in [12].

The current limits for the singlet-like VL lepton are shown in Fig. 3. We see that the \(\texttt{SR0}^{\textrm{tight}}_{\textrm{bveto}}\) gives the strongest bounds on the \(\textrm{BR}\left({E_{1}}\to{{Z^{\prime}}\mu}\right)\). Typically, \(\textrm{BR}\left({E_{1}}\to{{Z^{\prime}}\mu}\right)\sim 1\) when \(m_{\chi}>m_{E_{1}}>m_{Z^{\prime}}\), while \(\textrm{BR}\left({E_{1}}\to{{Z^{\prime}}\mu}\right)\sim\textrm{BR}\left({E_{1}}\to{\chi\mu}\right)\sim 0.5\) when \(m_{E_{1}}>m_{Z^{\prime}},m_{\chi}\). The limit is about \(1~\textrm{TeV}\) \((750~\textrm{GeV})\) nearly independent of the \({Z^{\prime}}\) mass when the branching fraction is 1 (0.5). When the branching fraction is 1, the limit for the doublet-like lepton is about 1.3 TeV.

4 SUMMARY

In this work, we constructed a model with a complete VL fourth family and a \(U(1)^{\prime}\) gauge symmetry. The anomaly in the muon anomalous magnetic moment is explained by the 1-loop diagrams mediated by the \({Z^{\prime}}\) boson and the VL leptons, and those in the rare \(B\) decays are explained by the tree-level \({Z^{\prime}}\) boson exchange. We searched for good-fit parameter points by a global \(\chi^{2}\) analysis, and we found plenty of points that can explain the anomalies. At these points, the other observables, such as lepton flavor violating decays and neutral meson mixing, are consistent with the current limits. We then proposed a novel possibility to detect signals with four muons or more at the LHC. By recasting the latest data, the current limit for the singlet-like (doublet-like) VL lepton is 1.0 (1.3) TeV when the \(\textrm{BR}\left({E}\to{{Z^{\prime}}\mu}\right)=1\).