Light $Z'$ boson from scalar boson decay at collider experiments in an $U(1)_{L_\mu - L_\tau}$ model

We study a model with $U(1)_{L_\mu - L_\tau}$ gauge symmetry which is spontaneously broken by a vacuum expectation value of a singlet scalar field. In this model, a light $Z'$ boson plays a role in explaining anomalous muon $g-2$ via one-loop effect and a CP-even scalar boson $\phi$ with $\sim \mathcal{O}(100)$ GeV mass appears associated with the symmetry breaking. We investigate experimental constraints for $U(1)_{L_\mu - L_\tau}$ gauge coupling, kinetic mixing, and mixing between the SM Higgs and $\phi$. Then collider physics is discussed considering $\phi$ production followed by decay process $\phi \to Z' Z'$ at the large hadron collider (LHC) and the international linear collider (ILC). In particular, we estimate discovery significance at the ILC taking into account relevant kinematical cut effects.


I. INTRODUCTION
The standard model (SM) of particle physics has been describing phenomena over the wide range of energy scale from eV to TeV scale. Despite of such enormous success, the anomalous magnetic moment of the muon, (g − 2) µ , shows a long-standing discrepancy between experimental observations [1,2] and theoretical predictions [3][4][5][6], where a µ = (g − 2) µ /2. This difference reaches to 3.6σ deviation from the prediction and seems not to be resolved within the SM. The on-going and forthcoming experiments will verify the discrepancy with high statistics, which will reduce the uncertainties by a factor of four [7,8]. Then, when the discrepancy is confirmed by the these experiments, it must be a firm evidence of physics beyond the SM.
Many extensions of the SM have been proposed to resolve the discrepancy so far. 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 (g − 2) µ can be explained by a new contribution from the associated gauge boson of the symmetry through loop diagrams. The L µ − L τ gauge symmetry is particularly interesting in this regard because it is anomaly free extension and can also explain the neutrino mass and mixings simultaneously [9,10].
In this model, it was shown in refs. [11][12][13] that the deviation of (g − 2) µ 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 a gauge boson with similar mass and gauge coupling can also explain the deficit of cosmic neutrino flux reported by IceCube collaboration [14][15][16][17][18]. Many experimental searches have been prepared and ongoing for such a light particles in meson decay experiment [19], beam dump experiment [20] and electron-positron collider experiment [21].
As mentioned above, the L µ − L τ 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 Scalar Lepton H ϕ L e L µ L τ e R µ R µ τ SU (2) L 2 1 2 2 2 1 1 1  [22,23]. In this paper, we study signatures for the scalar as well as the light gauge boson using Z ′ − Z ′ − φ vertex at the LHC experiment and the ILC experiment.
This paper is organized as follows. In section II, we briefly review the minimal gauged L µ − L τ model and give the partial decay widths of the scalar and gauge bosons. In section III, we show the allowed parameter space of the model. Then we show our results on the signatures of the scalar and the gauge boson production at the LHC and ILC experiments in section IV. Section V is devoted to the summary and discussion.

II. MODEL
We begin our discussions with reviewing a model with gauged U(1) Lµ−Lτ symmetry under which muon (µ) and tau (τ ) flavor leptons are charged among the SM leptons. As a minimal setup, we introduce a SM singlet scalar field ϕ to break the L µ −L τ symmetry spontaneously.
The gauge charge assignment for the lepton and scalar fields are given in Table I, and the quark sector is the same as that of the SM. In the table, L e , L µ , L τ and e R , µ R , τ R denote the left and right-handed leptons, and H denotes the SU(2) L doublet scalar field, respectively. The Lagrangian of the model is given by where L SM , J Z ′ and V represent the SM Lagrangian, the U(1) Lµ−Lτ current and the scalar potential, respectively. The gauge fields and its field strengths corresponding to U(1) Lµ−Lτ and U(1) Y are denoted by Z ′ and B. In Eq.(2), D µ = ∂ µ − ig ′ Z ′ µ is the covariant derivative, and g ′ and ǫ represent the new gauge coupling constant and the kinetic mixing parameter, respectively. In the following discussions, we assume that the quartic couplings of the scalar fields, λ H , λ ϕ and λ Hϕ , are positive to avoid runaway directions. In Eq.(4), µ 2 H and µ 2 φ are the tachyonic masses of H and ϕ.
The scalar fields H and ϕ can be expanded as where H + , A and a are massless Nambu-Goldstone bosons which should be absorbed by the gauge bosons W + , Z and Z ′ , whileH andφ represent the physical CP-even scalar bosons.
The VEVs of the scalar fields, v and v ϕ , are obtained from the stationary conditions Without loss of generality, the VEVs are taken to be real-positive by using the degree of freedom of the gauge symmetries to rotate the scalar fields. Inserting Eq.(5) into Eq.(4), the squared mass terms for CP-even scalar bosons are given by The above squared mass matrix can be diagonalized by an orthogonal matrix. The mass eigenvalues are given by and the corresponding mass eigenstates h and φ are obtained as where α is the mixing angle. When α ≪ 1, h is identified as the SM-like Higgs boson.
After the spontaneous breaking of the electroweak and L µ − L τ symmetries, the gauge bosons acquire masses. The neutral components of the gauge bosons mix each other due to the kinetic mixing while the charged ones remain the same as those of the SM. Assuming ǫ ≪ 1, the mass eigenvalues of the neutral components, Z 1,2,3 , are obtained after diagonalizing the mass term as well as the kinetic term, where m Z and θ W are the Z boson mass and the Weinberg angle in the SM, respectively, and The corresponding mass eigenstates of the gauge bosons are given by up to O(ǫ 2 ). Thus, Z 1 is the photon, and Z 2 and Z 3 are almost Z and Z ′ , respectively. We denote Z 1 and Z 2 as Z and Z ′ in the rest of this paper.
The Yukawa and gauge interactions of the SM fermions and φ in mass-basis are given by where m f and J EM represent the mass of the fermions f and the electromagnetic currents of the SM, and e and θ W are the electric charge of the proton and the Weinberg angle, respectively. In Eq. (13), the interactions between Z ′ and J EM are induced through the kinetic mixing. In the LHC and lepton collider experiments, the scalar φ can be mainly produced via the gluon fusion and associate Z production processes. One can see from Eq.(13) that the relevant interactions are proportional to the scalar mixing, sin α. Therefore the production cross section increases as sin α becomes larger.
For the SM-like Higgs boson, the gauge and scalar interactions in mass-basis are also obtained by inserting Eq.(9) into the Lagrangian. The relevant interactions in our discussions are given by where g hφφ is the constant given by There exist other gauge and scalar-self interactions involving h. However, those are negligible when the mixing angle α and the kinetic mixing parameter ǫ is much smaller than the unity.
Note that λ Hϕ is written in terms of α from Eq.(9). Therefore g hφφ becomes proportional to α when α is small enough.
In the end of this section, we show the decay widths of φ, Z ′ and h. As we will explain in the next section, we focus our discussions on the situation that the Z ′ gauge boson has a mass lighter than 2m µ , while the scalar boson φ has a mass of order 10-100 GeV. Thus, the φ can decay into Z ′ as well as the SM fermions and the gauge bosons. The partial decay widths of φ are given by where m Z,W ± are the mass of the gauge bosons, respectively. Here we have assumed final states are on-shell. It is important to mention that φ dominantly decays into Z ′ Z ′ when Z ′ mass is light since its partial decay width is enhanced by m 4 φ /m 4 Z ′ factor. The Z ′ boson can decay intoν µ,τ ν µ,τ or e + e − modes because m Z ′ < 2m µ . Then the partial decay widths of Z ′ are obtained as where we have ignored the neutrino masses and mixing. The branching ratio (BR) can be parametrized by the ratio of L µ − L τ gauge coupling and kinetic mixing parameter, ǫ/g ′ . We show BR(Z ′ → f f ) as a function of ǫ/g ′ in Figure 1 where red and blue curves respectively correspond toν µ,τ ν µ,τ and e + e − mode. The mass of Z ′ is fixed to 100 MeV, however the branching ratio is almost independent of the Z ′ mass when m Z ′ ≫ m e . It is seen in Fig. 1 that Z ′ mainly decays into neutrinos for ǫ/g ′ < 1. For later use, the branching ratio is about 0.07 for ǫ/g ′ = 1.
The SM-like Higgs boson can decay into not only Z ′ but also φ when m φ < m h /2. The partial widths of these decays are given by (21b) As we mentioned above, φ dominantly decays into Z ′ , and Z ′ mainly decays into neutrinos for ǫ/g ′ < 1. Therefore these decays are invisible. The branching ratio of the invisible decays in our model is given by where Γ SM is the total width of the Higgs boson in the SM 1 . When m φ ≥ m h /2, Γ h→φφ should be dropped in Eq. (22). The invisible decay of the Higgs boson has been searched at the LHC experiment in the production via gluon fusion [25], vector boson fusion [25][26][27][28], and in association with a vector boson [25,26,[28][29][30][31]. We employ BR(h → invisibles) ≤ 0.25 given in [28]. In Figure 2, the branching ratio is shown in m Z ′ -g ′ plane. The blue, green and red lines correspond to BR(h → invisibles) = 0.25, 0.05 and 0.001, respectively. The 1 Another invisible decay of the Higgs boson h → ZZ * → νννν, exists within the SM. The partial width of this decay is about 4.32 keV [24], and it is much smaller than the widths of h → Z ′ Z ′ /φφ in our parameter region. Thus, we have neglected this. scalar mass is taken as m φ = 30 GeV (solid) and m φ ≥ m h /2 (dashed), and the scalar mixing is fixed to sin α = 0.03 for reference. The SM-like Higgs mass and its total decay width is taken as 125 GeV [32] and 4.07 MeV [33], respectively. From the figure, we can see that g ′ should be smaller than 2 × 10 −3 for m Z ′ ≤ 2m µ , to avoid the upper bound from the LHC experiment. This region of g ′ is consistent with the favored region to resolve (g − 2) µ discrepancy.
As we mentioned in the introduction, the deviation of (g − 2) µ between the experimental observations and the theoretical prediction are within 2σ (3σ). The contribution from Z ′ to the anomalous magnetic moment is given by The favored region of the gauge coupling and the Z ′ mass to explain the deviation were studied in [45][46][47][48]. The region is summarized as The VEV of ϕ is estimated from Eqs. (25) and (26) as Since the mass of φ is roughly given by λ ϕ v ϕ , it is naturally expected that m φ is the same order of v ϕ . The most stringent bound on the kinetic mixing parameter is set by NA64 [19]. Based on the analysis in [47], the constraint from the meson decay is obtained by where ǫ MD is the upper bound in [19], which depends on m Z ′ . For m Z ′ = 100 (5) MeV, the favored region of ǫ is obtained as respectively.
On the other hand, the scalar mixing and the invisible Higgs decay branching ratio, BR invis , are also constrained by analysis of data from the LHC experiment [49,50] as In figure 3, we show the allowed region of sin α in m Z ′ -g ′ plane. In the left and right panels, m φ is taken as 30 GeV and larger than m h /2, respectively, and ǫ/g ′ = 1 is assumed. Note that for lighter m Z ′ , ǫ/g ′ = 1 is excluded by NA64. However, when we use ǫ/g ′ = 0.6, the (g − 2) µ favored region and the excluded region are slightly shifted upward in this case.
Therefore, the result does not change so much. In the following analysis, we fix m Z ′ = 100 MeV, ǫ/g ′ = 1 and sin α = 0.05, and discuss the observation possibilities at the LHC and the ILC collider experiments.

IMENTS
In this section, we discuss signature of φ and Z ′ in collider experiments; hadron collider such as the LHC and lepton collider such as the ILC. The scalar boson φ can be produced in collider experiments through mixing between the SM Higgs boson, and φ dominantly decays into Z ′ boson. We investigate possibility to search for the signature of φ and Z ′ in collider experiments.

A. Signatures at hadron collider
Here we discuss φ production processes and possibility to search for its signature at the LHC. The scalar boson can be produced by gluon fusion process gg → φ through mixing with the SM Higgs boson. The relevant effective interaction for the gluon fusion is given by [51] L φgg = α s 16π where G a µν is the field strength for gluon and This effective interaction is induced fromttφ coupling via the mixing effect where we take into account only top Yukawa coupling since the other contributions are subdominant. In Fig. 4, we show the production cross section estimated by MADGRAPH5 [52] implementing the effective interaction by use of FeynRules 2.0 [56], which is multiplied by scaling factor κ α ≡ (0.05/ sin α) 2 since the cross section is proportional to sin 2 α.
In principle, we can generate φ at the LHC with the cross section larger than ∼fb when scalar mixing is not very small and sizable events of pp → φ → Z ′ Z ′ → e + e + e − e − can be induced even if BR(Z ′ → e + e − ) is ∼ 0.07. However e + e − pair from Z ′ decay will be highly collimated when m Z ′ is lighter than GeV scale. Here we briefly estimate the degree of collimation; if Z ′ → e + e − decay system is boosted with velocity of where we assumed e ± direction before boost is z-direction and v Z ′ is perpendicular to the direction. Then the angle is ∼ 1 • for m Z ′ = 100 MeV and m φ = 50 GeV. Thus it is challenging to analyze signal with such a collimated e + e − [53,54]. Here we just show the distribution of transverse momentum of Z ′ in Fig. 5.
On the other hand if Z ′ decays into νν signature of φ includes missing transverse energy and it is difficult to reconstruct m φ at Hadron colliders. In that case lepton collider is more suitable to search for the signature.

B. Signatures at lepton collider
Here we discuss φ production processes and possibility to search for its signature at lepton colliders such as the ILC. In lepton collider experiments, φ can be produced by the processes such that e + e − → Zφ, e + e − → ννφ and e + e − → e + e − φ where the second process is W boson fusion and the third process is Z boson fusion; these processes are induced by the interactions in Eq. (13). In Fig. 6, we show the production cross sections for √ s = 250 GeV calculated by CalcHEP 3.6 [55] implementing relevant interactions, which is scaled by κ α = (0.05/ sin α) 2 factor. The figure shows that e + e − → Zφ mode gives the largest cross section for m φ 160 GeV in which the φ production is kinematically allowed. In our following analysis, we thus focus on the Zφ mode since cross sections for the other modes are small. Then we consider two cases; (1) Z decays into ℓ + ℓ − (ℓ = e, µ) and (2) Z decays into two jets where, in both cases, φ decays as φ → Z ′ Z ′ → νννν which is the dominant decay mode. Therefore our signals are for cases (1) and (2)  Hereafter we perform a simulation study of our signal and background (BG) processes in both cases (1) and (2) Here we discuss the "ℓ + ℓ − + / E" signal and corresponding BG events and estimate discovery significance applying relevant kinematical cuts. In this case we consider following BG processes: where the first process mainly comes from e + e − → ZZ/W + W − followed by leptonic decays of Z/W ± while the second process gives ℓ + ℓ − + / E evens via leptonic decay of τ ± . Signal and BG events are generated with basic cuts implemented in MADGRAPH/MADEVENT 5 as where p T denotes transverse momentum and η = 1/2 ln(tan θ/2) is the pseudo-rapidity with θ being the scattering angle in the laboratory frame.
We then investigate kinematic distributions for signal and BGs and efficiency of kinematical cutoff. Plots in Fig. 7 show ℓ + ℓ − invariant mass distributions where left-, middle-and right-panels correspond to events from signal, ℓ + ℓ − νν BG and τ + τ − BG with only basic cuts in Eq. (34). We find that the distribution for signal events shows clear peak at Z boson mass. On the other hand the distribution for ℓ + ℓ − νν BG has peak at Z mass and continuous region coming from e + e − → W + W − process. The distribution for τ + τ − BG has broad bump peaked around 80 GeV. To reduce BG events, we thus impose ℓ + ℓ − invariant mass cuts as m Z − 10 GeV < M ℓ + ℓ − < m Z + 10 GeV.
Furthermore we reconstruct the mass of φ using energy momentum conservation. The reconstructed mass is given by where E ℓ ± is energy of final state ℓ ± . Plots in Fig. 8 show the distribution of M rec φ for signal and BGs. We see that the mass of φ is indeed reconstructed giving clear peak. Note also that ℓ + ℓ − νν BG has peak at Z boson mass which comes from e + e − → ZZ process due to energy momentum conservation. Then we also impose kinematical cuts for M rec φ such that The Table II summarizes the effect of kinematical cuts to signal and BGs. We see that the number of events for BGs can be highly reduced by the M ℓ + ℓ − and M rec φ ℓ cuts while that of signal events does not change significantly. Note that number of BG events is large in the region M rec φ ℓ 80 GeV and it would be difficult to search for our signal if m φ is in the region.
Finally we estimate the discovery significance by where N S and N BG respectively denote the number of events for signal and total BG. The significances before and after kinematical cuts are shown in the last column of Table II. After imposing all cuts, we obtain discovery significance of 1.5(2.2) for m φ = 65(30) GeV with κ α = 1 corresponding to sin α = 0.05. Thus small scalar mixing as sin α = 0.05 will be constrained when mass of φ is as light as 30 GeV. Furthermore if sin α ∼ 0.1 we can get discovery significance larger than 5 since κ α ∼ 4. Note that more detailed kinematical cuts will improve the significance [61] but it is beyond the scope of this paper.

The case of jj + / E signal
Here we discuss the "jj + / E" signal and corresponding BG events and estimate discovery significance applying relevant kinematical cuts. In this case we consider following BG Only basic cuts (1.6 × 10 2 , 1.8 × 10 2 ) 1.1 × 10 6 1.9 × 10 5 (0.14, 0.16) processes: • e + e − → jjνν , where the first process mainly comes from e + e − → ZZ followed by Z decay into jets/neutrinos and the second process gives jj + / E events due to miss-identification of τ -jet as hadronic jets with missing energy. Signal and BG events are generated with basic cuts for jets in final states implemented in MADGRAPH/MADEVENT 5 as As in the ℓ + ℓ − + / E case, we investigate kinematical distributions for signal and BGs to find relevant kinematical cuts. Plots in Fig. 9 show distributions of invariant mass of two jets where left-, middle-and right-panels correspond to events from signal, jjνν BG and τ + τ − BG with only basic cuts in Eq. (39). The distribution for signal shows Z peak which is slightly broader than that in ℓ + ℓ − case above and the position of peak is slightly smaller than Z boson mass; this is due to the fact that jet energy resolution is worse than that of charged leptons. The jjνν BG case also shows distribution peaked around Z boson mass.
The distribution for τ + τ − BG shows broad bump peaked around 160 GeV. In reducing BG events, we thus impose jj invariant mass cuts such that We also reconstruct the mass of φ as in the case of charged lepton final state with energy momentum conservation. Similarly we obtain the reconstructed mass as where E j i is energy of a jet in final state j i . Plots in Fig. 10 show the distribution of M rec φ j for signal and BGs. We see that the reconstructed mass of φ tends to larger than actual value of m φ and peak for jjνν BG is also larger than m Z . This is due to energy loss of two jets due to initial/final state radiation which is stronger than the case of charged lepton + M rec φ j cut for m φ = 65 GeV (3.8 × 10 2 ,· · · ) 2.5 × 10 4 3.4 × 10 2 (2.4, · · · ) + M rec φ j cut for m φ = 30 GeV (· · · , 4.7 × 10 2 ) 1.7 × 10 3 19.
(· · · , 11) for m φ = 65(30) GeV. The Table III summarizes the effect of kinematical cuts to signal and BGs. We find that τ + τ − BG is highly suppressed by M jj and M rec φ j cuts, and main BG after cuts is jjνν one.
Finally we estimate the discovery significance using Eq. (38) which is shown in the last column of Table III. Significance tends to higher than that of ℓ + ℓ − + / E case; this is due to the facts that higher number of signal events by BR(Z → jj) > BR(Z → ℓ + ℓ − ) and e + e − → W + W − process does not contribute to jjνν final state. We then obtain significance much larger than 5 for m φ = 30 GeV with κ α = 1 corresponding to sin α = 0.05; S c ∼ 5 can be obtained with sin α = 0.03.

V. SUMMARY AND DISCUSSION
We have studied a model with U(1) Lµ−Lτ gauge symmetry which is spontaneously broken by a VEV of SM singlet scalar field with non-zero L µ − L τ charge. In this model Z ′ boson and new CP-even scalar boson φ are obtained after spontaneous symmetry breaking. Then we have focused on parameter region which can explain muon g − 2 by one-loop contribution where Z ′ boson propagates inside a loop, taking into account current experimental constraints. In the parameter region Z ′ mass range is 5 MeV m Z ′ 210 MeV, and mass of φ is typically O(100) GeV. We have also found that φ dominantly decays into Z ′ Z ′ mode and Z ′ decays into e + e − orν ℓ ν ℓ modes depending on the ratio between U(1) Lµ−Lτ gauge coupling constant and kinetic mixing parameter.
Then we have investigated signatures of φ production processes in collider experiments.
Firstly gluon fusion production of φ at the LHC has been discussed considering mixing between the SM Higgs boson and φ; the cross section is thus proportional to sin 2 α with mixing angle α. In principle we can obtain sizable number of events from pp → φ → Z ′ Z ′ followed by decay of Z ′ → e + e − even if Higgs-φ mixing is as small as sin α 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 φ production at e + e − collider such as the ILC. In e + e − collider, φ can be produced via e + e − → Zφ, W boson fusion and Z boson fusion processes through the mixing with the SM Higgs boson. Among them Zφ 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 φ mass can be reconstructed even if final state includes missing energy.
We have then considered the process e + e − → Zφ where φ decays into missing energy as φ → Z ′ Z ′ → 4ν since BR(Z ′ → νν) ≫ BR(Z ′ → e + e − ) in the parameter region to give sizable muon g −2. For Z boson decay, we have discussed two cases (1) Z → ℓ + ℓ − (ℓ = e, µ) and (2) Z → jj giving "ℓ + ℓ + + / E" and "jj + / E" signal events respectively. Numerical simulation study has been carried out for these cases generating signal events and the SM background events. 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 1.5(2.2) has been obtained for "ℓ + ℓ + + / E" signal when we take sin α = 0.05, m φ = 65(30) GeV and integrated luminosity of 3000 fb −1 . Furthermore the significance of 2.4 (11) has been obtained for "jj + / E" signal when we take sin α = 0.05 and m φ = 65(30) GeV, which is larger than the case with charged lepton final state. In addition, we can obtain larger significance for larger sin α although muon g − 2 tends to become smaller. Therefore we can search for the signal of φ at e + e − collider with sufficient integrated luminosity, and combining together with results from future muon g − 2 measurements our U(1) Lµ−Lτ model will be further tested. Note also that the significance would be improved by more sophisticated cuts and further analysis will be given elsewhere.