Searching for scalar boson decaying into light \(Z'\) boson at collider experiments in \(U(1)_{L_\mu  L_\tau }\) model
Abstract
We study a model with \(U(1)_{L_\mu  L_\tau }\) gauge symmetry and discuss collider searches for a scalar boson, which breaks \(U(1)_{L_\mu  L_\tau }\) symmetry spontaneously, decaying into light \(Z'\) gauge boson. In this model, the new gauge boson, \(Z'\), with a mass lighter than \({\mathcal {O}}(100)\) MeV, plays a role in explaining the anomalous magnetic moment of muon via oneloop contribution. For the gauge boson to have such a low mass, the scalar boson, \(\phi \) with \({\mathcal {O}}(100)\) GeV mass appears associated with the symmetry breaking. We investigate experimental constraints on \(U(1)_{L_\mu  L_\tau }\) gauge coupling, kinetic mixing, and mixing between the SM Higgs and \(\phi \). Then collider search is discussed considering \(\phi \) production followed by decay process \(\phi \rightarrow Z' Z'\) at the large hadron collider and the international linear collider. We also estimate discovery significance at the linear collider taking into account relevant kinematical cut effects.
1 Introduction
Many extensions of the SM have been proposed to resolve the discrepancy so far (See for a review [7]). Among them, one of the minimal extensions is to add a new U(1) gauge symmetry to the SM. When muon is charged under the symmetry, the deviation of \((g2)_\mu \) can be explained by a new contribution from the associated gauge boson of the symmetry through loop diagrams. The \(L_\mu  L_\tau \) gauge symmetry is particularly interesting in this regard because it is anomaly free extension and can also explain the neutrino mass and mixings simultaneously [10, 11, 12]. In this model, it was shown in Refs. [13, 14, 15] that the deviation of \((g2)_\mu \) can be resolved when the gauge boson mass is of order 10 MeV and the gauge coupling constant is of order \(10^{4}\). Such a light and weakly interacting gauge boson is still allowed from experimental searches performed in past. Interestingly, it was also shown that the gauge boson with similar mass and gauge coupling can also explain the deficit of cosmic neutrino flux reported by IceCube collaboration [16, 17, 18, 19, 20]. Many experimental searches have been prepared and ongoing for such a light particles in meson decay experiment [21], beam dump experiment [22] and electronpositron collider experiment [23]. Theoretical studies on search strategy at collider experiment are also proposed (see e.g. [24, 25, 26, 27] for \(L_\mu  L_\tau \) model ^{1}).
As mentioned above, the \(L_\mu  L_\tau \) gauge boson has a mass, hence the symmetry must be broken. This implies that at least one new complex scalar, which is singlet under the SM gauge group, should exist to break the symmetry and give a mass to the gauge boson. Then, from the gauge symmetry, there must exist an interaction of two gauge bosons and one real scalar by replacing the scalar field with its vacuum expectation value (VEV). Since this interaction is generated after the symmetry breaking, the confirmation of the interaction by experiments is a crucial to identify the model. The VEV of the scalar can be estimated as about 10–100 GeV from the gauge boson mass and the gauge coupling. Thus, naively one can expect that the physical CPeven scalar emerging after the symmetry breaking has a mass of the same order. Such a heavy scalar can not be directly searched at low energy experiments, and hence should be searched at high energy collider experiments, i.e. the Large Hadron Collider (LHC) experiment and future International Linear Collider (ILC) experiment [28, 29]. In this paper, we study signatures for the scalar as well as the light gauge boson using \(Z'\)\(Z'\)\(\phi \) vertex at the LHC experiment and ZZ\(\phi \) vertex at the ILC experiments.
Contents of scalar fields and their charge assignments under \(SU(2)_L\times U(1)_Y \times U(1)_{L_\mu L_\tau }\)
Scalar  Lepton  

H  \(\varphi \)  \(L_e\)  \(L_\mu \)  \(L_\tau \)  \(e_R\)  \(\mu _R\)  \(\tau _R\)  
\(SU(2)_L\)  \(\mathbf 2 \)  \(\mathbf 1 \)  2  \(\mathbf 2 \)  \(\mathbf 2 \)  \(\mathbf 1 \)  \(\mathbf 1 \)  1 
\(U(1)_Y\)  \(\frac{1}{2}\)  0  \(\frac{1}{2}\)  \(\frac{1}{2}\)  \(\frac{1}{2}\)  \(\)1  \(\)1  \(\)1 
\(U(1)_{L_\mu  L_\tau }\)  0  1  0  1  \(\)1  0  1  \(\)1 
2 Model
3 Allowed parameter space
In this section, we show the allowed parameter space of \(g'\), \(\epsilon \) and \(m_{Z'}\), \(\alpha \). The parameters of \(Z'\) are tightly constrained by experiments such as beam dump experiments [41, 42], meson decay experiments [21, 43, 44, 45, 46], neutrinoelectron scattering measurements [47], electronpositron collider experiment [48, 49], neutrino trident production process [50, 51]. A hadron collider experiment such as the LHC also constrains the gauge interaction for heavier \(Z'\) region [25, 26, 27] although we will not discuss such a heavy \(Z'\). The parameters can be further constrained by requiring that the \(Z'\) gauge boson gives enough contributions to \((g2)_\mu \).
4 Signature of extra scalar boson and \(Z'\) in collider experiments
In this section, we discuss signature of \(\phi \) and \(Z'\) in collider experiments; the LHC and the ILC. We consider the mass of \(\phi \) and \(Z'\) are \(O(10100)\) GeV and O(100) MeV, respectively. The scalar boson \(\phi \) can be produced in collider experiments through the mixing with the SM Higgs boson, and dominantly decays into \(Z'\) bosons. As we showed in the previous section, \(Z'\) dominantly decays into \(\nu \bar{\nu }\), and subdominantly into \(e^+ e^\) for \(\epsilon /g' < 1\). We investigate possibilities to search for the signature of \(\phi \) and \(Z'\) in collider experiments in this situation.
4.1 Signatures at the LHC
In the parameter space of our choice, the gauge boson \(Z'\) is mainly produced from \(\phi \) decay at the LHC because \(Z'\) interacts with quarks only through the kinetic mixing. The main production of \(\phi \) is gluon fusion through the mixing with the SM Higgs.
To identify the gauge and scalar bosons, \(Z'\) should decay into \(e^+e^\) because \(Z'\) and \(\phi \) are electrically neutral. However, \(e^+e^\) pair from \(Z'\) decay will be highly collimated due to lighter \(Z'\) mass than GeV scale. Here we estimate the degree of collimation; if \(Z' \rightarrow e^+ e^\) decay system is boosted with velocity of \(v_{Z'} \sim \sqrt{m_\phi ^2/4m_{Z'}^2}/(m_\phi /2)\) which is induced by decay of \(\phi \rightarrow Z' Z'\), the angle between \(e^+\) and \(e^\) is approximately \(\theta \sim \cos ^{1} (18m_{Z'}^2/m_\phi ^2)\) where we assumed \(e^\pm \) direction before boost is zdirection and \(\mathbf {v}_{Z'}\) is perpendicular to the direction. Then the angle is \(\sim 1^\circ \) for \(m_{Z'} = 100\) MeV and \(m_\phi = 50\) GeV. It is discussed in [60, 61] that reconstruction of such a collimated \(e^+e^\) pair is experimentally challenging due to angle resolution with the ATLAS detector. The reconstruction of \(e^+ e^\) pair is possible for \(m_{Z'} \ge 15\) GeV, which is already excluded for muon \((g2)\) to be explained. Even for \(\mu ^+ \mu ^\) pair, the reconstruction has been simulated only above \(m_{Z'} \ge 1\) GeV. A new analysis would be needed for the reconstruction of \(e^+\) and \(e^\) momenta. However such a new analysis is beyond the scope of this paper and we do not discuss here. From this fact, lepton colliders are more suitable to search for \(\phi \) in our parameter choice because it can use missing energy search due to the precise knowledge of the initial energy.
4.2 Signatures at the ILC
Hereafter we perform a simulation study of our signal and background (BG) processes in both cases (1) and (2); the events are generated via MADGRAPH/MADEVENT 5 [59], where the necessary Feynman rules and relevant parameters of the model are implemented by use of FeynRules 2.0 [63], the PYTHIA 6 [66] is applied to deal with hadronization effects, the initialstate radiation (ISR) and finalstate radiation (FSR) effects and the decays of the SM particles, and Delphes [67] is used for detector level simulation.
4.2.1 The case of Open image in new window signal

\(e^+ e^ \rightarrow \ell ^+ \ell ^ \nu {\bar{\nu }}\) ,

\(e^+ e^ \rightarrow \tau ^+ \tau ^+\),
The number of events for signal (\(N_S\)), BG (\(N_{BG}\)) and significance (\(S_{cl}\)) for RR polarization case after each cut where we have adopted \(m_\phi = (30, 65)\) GeV as reference values. The integrated luminosity is taken as 900 fb\(^{1}\), and \(N_S(S_{cl})\) is given by the products of scaling factor \(k_\alpha \) and the value for \(\kappa _\alpha =1\)
\(\kappa _\alpha N_{S}^{\kappa _\alpha =1}\); \(m_\phi = (65, 30)\) GeV  \(N_{BG}^{\ell ^+\ell ^\nu {\bar{\nu }}}\)  \(N_{BG}^{\tau \tau }\)  \(\kappa _\alpha S_{cl}^{\kappa _\alpha =1}\)  

Only basic cuts  (51., 53.)  \(7.7 \times 10^4\)  \(6.3 \times 10^4\)  (0.14, 0.14) 
\(+ M_{\ell ^+ \ell ^}\) cut  (48., 49.)  \(2.1 \times 10^4\)  \(1.3 \times 10^4\)  (0.25, 0.27) 
\(+ M_{\phi _\ell }^{rec}\) cut for \(m_\phi =65\) GeV  (42., \(\cdots \))  \(2.2 \times 10^2\)  \(1.3 \times 10^2\)  (2.2, \(\cdots \)) 
\(+ M_{\phi _\ell }^{rec}\) cut for \(m_\phi =30\) GeV  (\(\cdots \), 34.)  \(1.7 \times 10^2\)  14.  (\(\cdots \), 2.5) 
The number of events for signal (\(N_S\)), BG (\(N_{BG}\)) and significance (\(S_{cl}\)) for RR and LL polarizations with integrated luminosity of 900 fb\(^{1}\) each and for sum of events from two polarizaitons, where we show cases for \(m_{\phi } = 65(30)\) GeV with all kinematical cuts imposed
\(\kappa _\alpha N_{S}^{\kappa _\alpha =1}\); \(m_\phi = 65(30)\) GeV  \(N_{BG}^{\ell ^+\ell ^\nu {\bar{\nu }}}\)  \(N_{BG}^{\tau \tau }\)  \(\kappa _\alpha S_{cl}^{\kappa _\alpha =1}\)  

RR  42.(34.)  \(2.2(1.7) \times 10^2\)  \(1.3(0.14)\times 10^2\)  2.2(2.5) 
LL  53.(47.)  \(4.7(1.7) \times 10^3\)  \(1.6(0.15) \times 10^2 \)  0.75(1.1) 
\(LL + RR\)  95.(81.)  \(4.9(1.9) \times 10^3\)  \(2.9(0.29) \times 10^2\)  1.3(1.8) 
4.2.2 The case of Open image in new window signal

\(e^+ e^ \rightarrow j j \nu {\bar{\nu }}\) ,

\(e^+ e^ \rightarrow \tau ^+ \tau ^+\),
The number of events for signal (\(N_S\)), BG (\(N_{BG}\)) and significance (\(S_{cl}\)) for RR polarization after each cut where the setting is the same as Table. 2
\(\kappa _\alpha N_{S}^{\kappa _\alpha =1}\); \(m_\phi = (65, 30)\) GeV  \(N_{BG}^{jj \nu {\bar{\nu }}}\)  \(N_{BG}^{\tau \tau }\)  \(\kappa _\alpha S_{cl}^{\kappa _\alpha =1}\)  

Only basic cuts  (\(3.8 \times 10^2\), \(1.2 \times 10^3\))  \(1.1 \times 10^5\)  \(6.1 \times 10^5\)  (0.45, 0.46) 
\(+ M_{jj}\) cut  (\(2.9 \times 10^2\), \(9.3 \times 10^2\))  \(8.0 \times 10^4\)  \(3.0 \times 10^4\)  (0.88, 1.1) 
\(+ M_{\phi _j}^{rec}\) cut for \(m_\phi =65\) GeV  (\(1.3 \times 10^2\),\(\cdots \))  \(5.7 \times 10^3\)  \(1.3 \times 10^2\)  (1.6, \(\cdots \)) 
\(+ M_{\phi _j}^{rec}\) cut for \(m_\phi =30\) GeV  (\(\cdots \), \(1.5 \times 10^2\))  \(3.3 \times 10^2\)  6.4  (\(\cdots \), 8.3) 
The number of events for signal (\(N_S\)), BG (\(N_{BG}\)) and significance (\(S_{cl}\)) for RR and LL polarizations with integrated luminosity of 900 fb\(^{1}\) each and for sum of events from two polarizaitons, where we show cases for \(m_{\phi } = 65(30)\) GeV with all kinematical cuts imposed
\(\kappa _\alpha N_{S}^{\kappa _\alpha =1}\); \(m_\phi = 65(30)\) GeV  \(N_{BG}^{jj \nu {\bar{\nu }}}\)  \(N_{BG}^{\tau \tau }\)  \(\kappa _\alpha S_{cl}^{\kappa _\alpha =1}\)  

RR  \(1.3 (1.5) \times 10^2\)  \(5.6 (0.33) \times 10^3\)  \(1.3 (0.064) \times 10^2\)  1.6(8.3) 
LL  \(1.6(1.9) \times 10^2\)  \(1.3 (0.085) \times 10^4\)  \(2.0 (0.13) \times 10^2 \)  1.4(6.5) 
\(LL + RR\)  \(2.9 (3.4) \times 10^2\)  \(1.9 (0.12) \times 10^4\)  \(3.3 (0.19) \times 10^2\)  2.1(9.7) 
Before closing this section, let us discuss the potential of the other lepton colliders and possibility of testing scalar mixing in future Higgs measurement. In addition to the ILC, the CEPC [70] and FCCee [71, 72] can investigate our scenario; the CEPC at \(\sqrt{s} = 240\) GeV can provide data with integrated luminosity of 5 ab\(^{1}\) while at the FCCee integrated luminosity can be 10(5) ab\(^{1}\) for \(\sqrt{s} = 160(\sim 250)\) GeV and that of 1.5 ab\(^{1}\) is possible for \(\sqrt{s}= 350\) GeV. Then these experiments also have the potential to find the signature of our model which would give similar significance as our analysis since the energy and integrated luminosity are not significantly different from the case of the ILC. Thus combining the analysis of these experiments we can further improve the test of our model. Moreover the lepton colliders can significantly improve measurements of the SM Higgs coupling which can constrain the scalar mixing. The couplings of hZZ interaction can be measured with the most strong sensitivity of \(\sim 0.1 \%\) error and the other coupling can be also measured with few \(\%\) error in each future lepton colliders [29, 70, 71]. In our case, the SM Higgs coupling is given by \(\cos \alpha \times C_{{h VV/ h{\bar{f}} f}}^{SM}\) where \(C_{hVV/ h{\bar{f}} f}^{SM}\) is the SM Higgs coupling with vector bosons/fermions. Thus divination from the SM is given by \(1  \cos \alpha \simeq 0.0013 \times (\sin \alpha /0.05)^2\) which would be tested by hZZ coupling measurement. The more stringent constraint can be obtained from future measurement of invisible decay branching ratio of the SM Higgs. For example, the ILC at \(\sqrt{s}=250\) GeV with integrated luminosity of 2 ab\(^{1}\) can explore the branching ratio up to \(0.32 \%\) [29]. Therefore, comparing with Fig. 2, wide parameter region can be explored which will be good complimentary test of our model.
5 Summary and discussion
We have studied a model with \(U(1)_{L_\mu  L_\tau }\) gauge symmetry which is spontaneously broken by a VEV of SM singlet scalar field with nonzero \(L_\mu  L_\tau \) charge. In this model \(Z'\) boson and new CPeven scalar boson \(\phi \) are obtained after spontaneous symmetry breaking. Then we have focused on parameter region which can explain muon \(g2\) by oneloop contribution where \(Z'\) boson propagates inside a loop, taking into account current experimental constraints. In the parameter region \(Z'\) mass range is 5 MeV \(\lesssim m_{Z'} \lesssim \) 210 MeV, and mass of \(\phi \) is typically \({\mathcal {O}}(100)\) GeV. We have also found that \(\phi \) dominantly decays into \(Z'Z'\) mode and \(Z'\) decays into \(e^+e^\) or \({\bar{\nu }}_\ell \nu _\ell \) modes depending on the ratio between \(U(1)_{L_\mu  L_\tau }\) gauge coupling constant and kinetic mixing parameter.
Then we have investigated signatures of \(\phi \) production processes in collider experiments. Firstly gluon fusion production of \(\phi \) at the LHC has been discussed considering mixing between the SM Higgs boson and \(\phi \); the cross section is thus proportional to \(\sin ^2 \alpha \) with mixing angle \(\alpha \). In principle we can obtain sizable number of events from \(pp \rightarrow \phi \rightarrow Z' Z'\) followed by decay of \(Z' \rightarrow e^+ e^\) even if Higgs\(\phi \) mixing is as small as \(\sin \alpha \lesssim 0.1\). However \(e^+e^\) pair from light \(Z'\) decay is highly collimated and it is very challenging to analyze the signal events at the LHC requiring improved technology.
Secondly we have investigated \(\phi \) production at \(e^+ e^\) collider such as the ILC. In \(e^+ e^\) collider, \(\phi \) can be produced via \(e^+e^ \rightarrow Z \phi \), W boson fusion and Z boson fusion processes through the mixing with the SM Higgs boson. Among them \(Z \phi \) mode can give the largest cross section if kinematically allowed and we have focused on the process. One advantage of \(e^+e^\) collider compared with hadron colliders is that we can use energy momentum conservation and \(\phi \) mass can be reconstructed even if final state includes missing energy. In addition, we can use polarized electron/positron beam at the ILC experiment. We have then considered the process \(e^+e^ \rightarrow Z \phi \) where \(\phi \) decays into missing energy as \(\phi \rightarrow Z' Z' \rightarrow 4 \nu \) since \(BR(Z' \rightarrow \nu {\bar{\nu }}) \gg BR(Z' \rightarrow e^+ e^)\) in the parameter region to give sizable muon \(g2\). For Z boson decay, we have discussed two cases (1) \(Z \rightarrow \ell ^+ \ell ^ (\ell = e, \mu )\) and (2) \(Z \rightarrow j j\) giving “ Open image in new window ” and “ Open image in new window ” signal events respectively. Numerical simulation study has been carried out for these cases generating signal events and the SM background events. In our analysis, we have applied two polarization case in which \((e^, e^+)\) beams are polarized as \(( 80 \%, + 30 \%)\) and \((+ 80 \%,  30 \%)\) denoted by LL and RR polarizations respectively. We have investigated relevant kinematical cuts to reduce the backgrounds showing corresponding distributions. Finally we have estimated discovery significance for our signal taking into account the effects of kinematical cuts. The significance of 2.2(2.5) has been obtained for “ Open image in new window ” signal when we take \(\sin \alpha =0.05\), \(m_\phi =65(30)\) GeV and integrated luminosity of 900 fb\(^{1}\) for RR polarization. Remarkably, we have the largest significance from RR polarization which is even larger than sum of LL and RR events since BG from \(e^+ e^ \rightarrow W^+ W^ \rightarrow \ell ^+ \ell ^ \nu {\bar{\nu }}\) process is suppressed in RR polarization. Furthermore the significance of 2.4(9.7) has been obtained for ” Open image in new window ” signal when we take \(\sin \alpha =0.05\) and \(m_\phi =65(30)\) GeV, which is larger than the case with charged lepton final state. In this case, we have find the largest significance can be obtained by simply summing up events from events LL and RR polarization. In addition, we can obtain larger significance for larger \(\sin \alpha \) although muon \(g2\) tends to become smaller. Therefore we can search for the signal of \(\phi \) at \(e^+e^\) collider with sufficient integrated luminosity, and combining together with results from future muon \(g2\) measurements our \(U(1)_{L_{\mu } L_{\tau }}\) model will be further tested. Note also that the significance would be improved by more sophisticated cuts and further analysis will be given elsewhere. For the last comment, we discuss displaced vertex of \(Z'\) decay into \(e^+ e^\). From Eq. (20), the order of the lifetime can be estimated as \(\tau _{Z'} \simeq 24 \pi /(g'^2 m_{Z'}) \sim 4 \times 10^{14}\) sec., where we assumed \(g' = 10^{4}\) and \(m_{Z'} = 100\) MeV. The decay length is \(c \tau _{Z'} \sim 1\) cm which is comparable with the radius of an innermost vertex tracker at the ILC. Therefore displaced vertices of \(Z'\) decaying into \(e^+ e^\) might be measured if enough number of \(Z'\) is produced.
Footnotes
 1.
In these analyses, \(Z'\) mass is considered to be \({\mathcal {O}}(10)\)–\({\mathcal {O}}(100)\) GeV and \(Z'\) can decay into charged leptons \(\mu ^+\mu ^(\tau ^+\tau ^)\) providing four charged lepton signals.
 2.
Then \(Z'\) interaction is flavor diagonal and Kmeson and Bmeson physics do not give significant constraints to the \(Z'\) coupling and mass.
 3.
Another invisible decay of the Higgs boson \(h \rightarrow ZZ^*\rightarrow \nu \nu \bar{\nu } \bar{\nu }\), exists within the SM. The partial width of this decay is about 4.32 keV [31], and it is much smaller than the widths of \(h \rightarrow Z'Z'/\phi \phi \) in our parameter region. Thus, we have neglected this.
Notes
Acknowledgements
This work is supported by JSPS KAKENHI Grants No. 15K17654 and 18K03651 (T.S.). The authors would like to thank Hideki Okawa and Shinichi kawada for the private discussion.
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