Probing the $L_\mu-L_\tau$ gauge boson at electron colliders

We investigate the minimal $U(1)_{L_\mu-L_\tau}$ model with extra heavy vector-like leptons or charged scalars. By studying the kinetic mixing between $U(1)_{L_\mu-L_\tau}$ gauge boson $Z^\prime$ and standard model photon, which is absent at tree level and will arise at one loop level due to $\mu$, $\tau$ and new heavy charged leptons or scalars, the interesting behavior is shown. It can provide possibility for visible signatures of new heavy particles. We propose to search for $Z^\prime$ at electron collider experiments, such as Belle II, BESIII and future Super Tau Charm Factory (STCF), using the monophoton final state. The parameter space of $Z^\prime$ is probed, and scanned by its gauge coupling constant $g_{Z^\prime}$ and mass $m_{Z^\prime}$. We find that electron colliders have sensitivity to the previously unexplored parameter space for $Z^\prime$ with MeV-GeV mass. Future STCF experiments with $\sqrt s=2-7$ GeV can exclude the anomalous muon magnetic moment favored area when $m_{Z^\prime}<5$ GeV with the luminosity of 30 ab$^{-1}$. For $m_{Z^\prime}<2m_\mu$, $g_{Z^\prime}$ can be down to $4.2\times 10^{-5}$ at 2 GeV STCF.


INTRODUCTION
The standard model (SM) of particle physics is a successful and highlypredictive theory of fundamental particles and interactions, but fails to explain many phenomena, including neutrino mass, baryon asymmetry of the universe, presence of dark matter (DM) and dark energy, among others. It implies that SM is only a low-energy approximation of the more fundamental theory; extensions of SM are strongly required.
Since Z can directly couple to muon, related searches for Z have been performed with the production of µ + µ − Z at collider experiments, including BaBar [27] and Belle II [28] at electron colliders and CMS [29] at hadron collider. Subsequently, Z decaying to muon-pair is considered at BaBar and CMS experiments, and invisible decay of Z is considered at Belle II. Phenomenally, Ref. [30] investegated the sensitivity on Z at Belle II with the planned target luminosity of 50 ab −1 in the channel of e + e − → µ + µ − Z , Z → INV; Refs. [31][32][33][34] proposed the search for Z at Belle II using the monophoton process e + e − → γZ , Z → invisible, which depends on the kinetic mixing between the SM photon and Z .
In this work, we investigate the γ − Z kinetic mixing in the minimal U (1) Lµ−Lτ with extra heavy vectorlike leptons or charged scalars. Then we propose to search for L µ − L τ gauge boson Z at electron collider experiments, such as Belle II, BESIII and future Super Tau Charm Factory (STCF), using the monophoton final state. Belle II is an asymmetric detector and located at SuperKEKB which collides 7 GeV electrons with 4 GeV positrons. SuperKEKB has a largest instantaneous luminosity of 8 × 10 35 cm −2 s −1 [35]. The ambitious goal of SuperKEKB is to accumulate an integrated luminosity of 50 ab −1 with 8-year data takings [35]. The BESIII detector is symmetric and operated on the BEPCII with the beam energy ranging from 1.0 GeV to 2.3 GeV and a peak luminosity of 10 33 cm −2 s −1 [36]. STCF is a proposed symmetric detector experiment which collides electron with positron in the range of center-of-mass energies from 2.0 to 7.0 GeV with the peak luminosity O(10 35 ) cm −2 s −1 at 4 GeV [37][38][39].
The rest paper is organized as follows: First, we introduce the U (1) Lµ−Lτ models and discuss their phenomenological features. Then, we calculate the cross sections of the signal and the backgrounds and analysis to improve the significance by appropriate event cuts at three different electron colliders operated at the GeV scale: BelleII, BESIII and STCF. The sensitivities for Z at these experiments are also investigated. Finally, a short summary and discussions are given.
We extend the SM with a new U (1) gauge symmetry, U (1) Lµ−Lτ , where leptons of the second and third generation couple to the additional U (1) Lµ−Lτ gauge boson Z with equal and opposite charge. The new leptonic gauge interactions can be given as (1) where g Z is gauge coupling constant.
In the minimal U (1) Lµ−Lτ model, the kinetic mixing between the Z and photon is absent at the tree level. Nevertheless, because µ and τ are both charged under the electromagnetic U (1) and U (1) Lµ−Lτ , there exists an unavoidable kinetic mixing at one loop level, which can appear as [32] Here e is the electromagnetic charge, m τ and m µ are the masses of tau and muon leptons, q is the transferred momentum.
For large momentum transfer q 2 m 2 τ , this mixing is power suppressed by 1/q 2 , whereas for low momentum transfer q 2 ∼ 0 m 2 µ , the mixing tends to be a constant which seems like the dark photon model. We add two extra singlet vectorlike leptons (L 1 , L 2 ) in the U (1) Lµ−Lτ extension of the SM, which are charged under U (1) Lµ−Lτ opposite in sign similar as the µ and τ , and have electric charge of e [33]. Since we mainly focus on the gauge kinetic mixing, we would not provide much details of the model here. In this model, due to the leptons inside the loop, the kinetic mixing of γ and Z can be derived as Here m L1 , m L2 are the masses of L 1 and L 2 . When the momentum transfer q 2 m L1/L2 , which is considered in this work, the mixing can be simplified as where r = m L2 /m L1 is the mass ratio of L 1 and L 2 .
The U (1)L µ−Lτ model with extra heavy charged scalars In the U (1) Lµ−Lτ extension of the SM, we add two extra scalars (S 1 , S 2 ) with electric charge of e and charged under U (1) Lµ−Lτ opposite in sign [34]. Similarly, due to charged leptons and extra scalars contributions induced at one-loop level, the γ − Z kinetic mixing can be given as Here m S1 and m S2 are the masses of extra charged scalars (S 1 and S 2 ). We mainly focus on the gauge kinetic mixing, thus much details of the model are not provided here.
In this work, we consider the momentum transfer always q 2 m S1/S2 , thus the mixing can be also written as ε HCS (q 2 , r) = ε min (q 2 ) + eg Z 24π 2 ln r, where r = m S2 /m S1 is the mass ratio of S 2 and S 1 . In Fig.1, we present the square of the kinetic mixing |ε HVL,HCS /g Z | 2 as a function of the momentum transfer |q| with r = 0.1, 1 and 10. The horizontal dotted lines are the same situations but for the case of ε(q 2 = 0), which are shown as a comparison. When r = 1, the contribution for the kinetic mixing due to additinal leptons or scalars vanishes, and the results will become same as those in the minimal U (1) Lµ−Lτ model, i.e., ε HVL (q 2 , 1) = ε HCS (q 2 , 1) = ε min (q 2 ). In the minimal U (1) Lµ−Lτ model, |ε/g Z | 2 has two peaks at the position of |q| = m µ and |q| = m τ , and drops quickly with the increment of |q| when |q| > m τ . This feature distinguishes the phenomenology of the U (1) Lµ−Lτ model from the dark photon models with a constant value of the kinetic mixing.
We also present the dependence of the kinetic mixing ratio R = |ε HVL/HCS | 2 /|ε min | 2 between the U (1) Lµ−Lτ model with two singlet vectorlike leptons or with two charged scalars and the minimal U (1) Lµ−Lτ model on the mass ratio r in Fig.2. There we consider five typical momentum transfers |q| = 0.1 GeV, 1 GeV, 10 GeV, 2m µ , 2m τ . It can be seen that, the additional lepton or scalar contributions could be significant, and the results are distinctly different from those of the minimal U (1) Lµ−Lτ model. Though the additional leptons and scalars cannot be detected directly due to their heavy mass, they can provide significant contributions to the kinetic mixing.

Decay modes of Z
Since the Z direct couples with the leptons of second and third generation, it can decay into a pair of neutrinos, and also may decay into muon and tau leptons if kinematic allowed. In addition, since Z provides possible scenarios of dark matter, there can be the channel Z → χχ. The decay widths of Z are given by, where = {µ, τ }, g D is the coupling constant of the Z with dark matter, and g D g Z is assumed. We ignore the channel Z → e + e − since it is suppressed by the kinetic mixing. Since neutrinos and dark matter are invisible at particle detectors, we take the Z invisible decay as Γ(Z → INV) = Γ(Z → νν) + Γ(Z → χχ), whose decay ratio can be expressed as

EXISTING CONSTRAINTS
In this section, we summarize the existing constraints relevant to the parameter regions we are interested for the minimal U (1) Lµ−Lτ model from various experiments as follows: • Muon anomalous magnetic moment. The significant discrepancy between the experimental measurement and the SM prediction in the magnetic moment of the muon remains one of the largest anomalies in particle physics [40]: where the errors are from experiment and theory prediction, respectively. We require the contribution in Eq. (12) to be within 2σ that leads to 103 ∆a Z µ × 10 11 420.
The minimal U (1) Lµ−Lτ model, was first introduced to address the discrepancy, which can provide a new interaction with muons. An extra contribution to a µ arises solely from a one-loop diagram involving Z , which can be giving by The parameter region on which the Z contribution in the minimal L µ − L τ model resolves the discrepancy in the muon anomalous magnetic moment at 2σ is indicated with the red band in Fig. 3.
• Neutrino trident production. The neutrino trident production is a muon neutrino scattering off the Coulomb field of a target nucleus (N ), producing two muons in the final state, νN → νN µ + µ − . Besides the SM Z boson, in the U (1) Lµ−Lτ model, the Z boson can also contribute to this process, which can offer a sensitive search for the light Z    boson [41,42]. The measurements for the cross section have been reported by CCFR, which obtain the result σ CCFR /σ SM = 0.82 ± 0.28. The bound is depicted in Fig. 3 and taken from Ref. [41].
• Z production associated with muon pair.
Via the direct coupling to µ, Z can be produced at e + e − colliders in the process e + e − → µ + µ − Z . Babar experiment has reported the bounds using 514 fb −1 data collected in the reaction e + e − → µ + µ − Z , Z → µ + µ − for m Z > 2m µ [27]. Recently, Belle II experiment perform the first searches for the invisble decay of a Z in the process e + e − → µ + µ − Z , Z → INV using 276 pb −1 collected [28], which can touch the region of m Z < 2m µ .
• Z production associated with SM photon.
At e + e − colliders, the Z boson can also be produced associated with SM photon via the kinetic mixing in the process e + e − → γZ [48]. The search for invisible decays of dark photon has been preformed at BaBar experiment using the single-photon events with 53 fb −1 data. We translate the constraints for dark photon to U (1) Lµ−Lτ gauge boson Z using where ε DP is the photon and dark photon kinetic mixing parameter in the dark photon model, and ε is the γ − Z kinetic mixing in the U (1) Lµ−Lτ model.
In Fig. 3, we asume Z does not decay into dark sector, i.e., Γ(Z → INV) = Γ(Z → νν). The BR(Z → INV) 1 cases are also shown as dotted line for a visual display. Taking the constraints above into account, a narrow window of the m Z − g Z parameter region in the minimal U (1) Lµ−Lτ model desired by the muon anomalous magnetic moment,  BaBar γ +INV (g− 2)µ ±2σ Figure 3: Summary for the m Z − g Z parameter space of the mininal U (1)L µ −Lτ model, where Z has no additional decay channel to dark sector. The shaded regions show the exisiting bounds excluded by CCFR experiment in neutrino trident production [41], by Borexino detector in neutrino-electron scattering [46], by BaBar in the reactions e + e − → µ + µ − Z , Z → µ + µ − with 514 fb −1 data [27] and e + e − → γZ , Z → INV with 53 fb −1 data [48], and by Belle II in the process e + e − → µ + µ − Z , Z → INV with 276 pb −1 data [28]. The dotted lines indicate BR(Z → INV) 1 cases. The red band indicate the allowed region at 2σ from the experimental measurements of muon magnetic momentum.
At the electron colliders, the production of Z can be associated with a SM photon through the kinetic mixing in the process e + e − → γZ , whose diagrams are shown in Fig.4. Subsequently, the produced Z boson can decay into charged leptons, a pair of neutrinos or light dark matter. In this paper, we focus on the Z invisible decay channel Z → INV, including Z → νν and Z → χχ, to probe Z boson via the monophoton searches e + e − → γZ → γ + INV at electron colliders. We assume that the decay width of the Z is negligible compared to the experimental resolution, which justifies the use of the narrow width approximation. In the monophoton signature at electron colliders, the major backgrounds (BGs) from SM contain two types: irreducible and reducible BG. The irreducible monophoton BG comes from the process e + e − → ννγ , where ν is the three neutrinos. The reducible monophoton BG arises from the electromagnetic processes e + e − → γ + / X, where / X denotes other visible particles but undetected due to the limitations of the detector acceptance. We discuss the reducible BG in detail later for each experiment, since it strongly depends on the angular coverage of the detectors.
The differential cross section for an on-shell Z and a photon production process e + e − → γZ is [49] where α is the fine structure constant, z γ ≡ cos θ γ with θ γ being the relative angle between the electron beam axis and the photon momentum in the center-of-mass (CM) frame, s is the square of the CM energy, m Z is the mass of the U (1) Lµ−Lτ gauge boson. The photon energy E γ in the CM frame is related to the Z mass as The cross section after integrating the polar angle θ γ is given as [49] where

BELLE II
At Belle II, photons and electrons can be detected in the Electromagnetic Calorimeter (ECL), which is made up of three segments: forward endcap with 12.4 • < θ < 31.4 • , barrel with 32.2 • < θ < 128.7 • , and backward endcap 130.7 • < θ < 155.1 • in the lab frame [35]. At Belle II, the reducible BG for monophoton singnature consists of two major parts: one is mainly due to the lack of polar angle coverage of the ECL near the beam directions, which is referred to as the "bBG"; the other one is mainly due to the gaps between the three segments in the ECL detector, which is referred to as the "gBG".
The bBG comes from the electromagnetic processes e + e − → γ + / X, manily including e + e − → / γ / γγ and e + e − → / e + / e − γ , where all the other final state particles except the detected photon are emitted along the beam directions with θ > 155.1 • or θ < 12.4 • in the lab frame. At Belle II, we adopt the detector cuts for the final detected photon (hereafter the "pre-selection cuts"): 12.4 • < θ γ < 155.1 • in the lab frame.
In Fig.5, we show the production rates of the process e + e − → γZ in U (1) Lµ−Lτ models after the "preselection cuts" for the photon at Belle II with √ s = 10.58 GeV. The dotted lines correspond to the case of constant ε(q 2 = 0), which are shown as a comparison. We can see that, with constant ε(q 2 = 0), the cross sections all increase with the increment of the mass of Z . In the minimal U (1) Lµ−Lτ model, the production rates for the process e + e − → γZ at Belle II generally drop but exist two peaks at the positions of m µ and m τ when m Z < 8.5 GeV, while raise at the tail of the plotted region.
For the Belle II detector, which is asymmetric, the maximum energy of the monophoton events in the bBG in the CM frame, E m γ , is given by [50] (if not exceeding √ s/2) where all angles are given in the CM frame, and A = (sin θ 1 − sin θ 2 )/(cos θ 1 − cos θ 2 ), with θ 1 and θ 2 being the polar angles corresponding to the edges of the ECL detector. In order to remove the above bBG, we use the detector cut E γ > E m γ (hereafter the "bBG cuts") for the final monophoton .
The gBG for the monophoton singnature have been simulated in the Ref. [35] to search for dark photons decaying into light dark matter. The projetced upper limits on the kinetic mixing of dark photon and SM photon ε for a 20 fb −1 Belle II dataset are present there. The constranint for the kinetic mixing between U (1) Lµ−Lτ gauge boson and SM photon ε can be translated from the dark photon using Eq. (15). We scale the expected sensitivity S(g Z ) to the planned full of integrated luminosity of 50 ab −1 at Belle II using S(g Z ) ∝ 4 √ L. Then the corresponding constraint based on the simulation in Ref. [35] from 20 fb −1 to 50 ab −1 can be simply projected by a factor of 4 50/ab 20/fb , which is present in Fig.6 and the invisible decay ratio Br(Z → INV) 1 is assumed. It is shown that the sensitivity for g Z at Belle-II experiment with 50 ab −1 via monophoton searches is expected to be worse in the minimal U We further carry out an analysis without gBG taking into account, to compare with other experiments in which detailed simulations with gBG are not available. We use the "bBG cuts" to remove the reducible BG events; this momentum the BG events survived the "bBG cuts" come from irreducible BG without gBG considered. Since the energy of the final photon in the signal process is related to m Z , in addiction to the "bBG cuts", we select final photon in the energy window of |E γ − (s − m 2 Z )/(2 √ s)| < σ E /2 (hereafter the "optimized cut") to enhance the discovery sensitivity, where σ E is detector energy resolution for the photon. At Belle II, σ E /E = 4%(1.6%) at 0.1 (8) GeV [35] and we take σ E = 128 MeV conservatively. In Fig. 6, we present the expected 95% confidence level (C.L.) exclusion limits on g Z by considering the irreducible BG only after "optimized cut", which is labeled as Belle-II . We define χ 2 (ε) ≡ S 2 /(S + B) [51], where S (B) is the number of events in the signal (BG) processes. The 95% C.L. upper bound on g Z is obtained by solving χ 2 ( 95 ) − χ 2 (0) = 2.71, and assuming photon detection efficiency as 95% [35]. One can see that if we don't consider the "gBG" and apply the "optimized cut", the Belle II experiment with 50 ab −1 via monophoton searches is expected to be sensitive to the parameter region with m Z 1.2 GeV and g Z 4 × 10 −4 in the minimal L µ − L τ model, which can be improved by almost 1 order of magnitude comparing with considering the "gBG".
Belle II, s =10.58 GeV  Belle II 50/ab Belle II 50/ab r =0.1 r =1 r =10 Figure 6: Sensitivity limit for g Z at Belle-II experiment with 50 ab −1 to search for dark photon decaying into light dark matter based on the simulation in Ref. [35], lablled as "Belle II", red color. The expected 95% C.L. exclusion limits on g Z via monophoton searches at 50 ab −1 Belle-II with gBG omitted after "optimized cut", labeled as Belle-II , black color. For Lµ − Lτ model with extra heavy vector-like leptons (Left) or charged scalars (Right) in the cases of r = 0.1 (dashed), 1 (solid) and 10 (dotted).

BESIII AND STCF
At BESIII and STCF, for the final state photons, we adopt the "preselection cuts" by BESIII Collaboration [52]: E γ > 25 MeV with | cos θ| < 0.8 or E γ > 50 MeV with 0.86 < | cos θ| < 0.92. In Fig. 7, we present the cross section of the the process e + e − → γZ at BESIII and STCF with √ s = 4 GeV in U (1) Lµ−Lτ models after the "pre-selection cuts". The dotted lines correspond to the case of constant ε(q 2 = 0), which are shown as a comparison. One can see that, the cross section always increases for larger m Z in U (1) Lµ−Lτ models with extra heavy leptons or scalars in the case of r = 0.1, while there is a twist near m Z = 2m τ in the case of r = 1 and r = 10.
At BESIII and STCF, which are symmetric, the maximum energy of the monophoton events in the bBG in the CM frame, E m γ , is given by [53] where cos θ b is the polar angle corresponding to the edge of the detector. Taking into account the coverage of MDC, EMC, and TOF, we have cos θ b = 0.95 at the BESIII [54]. We further demand E γ > E m γ for the final monophoton to remove the reducible BG (hereafter the  Figure 7: The cross sections of the process e + e − → γZ at BESIII or STCF with √ s = 10.58 GeV after the "pre-selection cuts" for Lµ − Lτ model with extra heavy vector-like leptons (Left) or charged scalars (Right). The horizontal dotted lines are the same situations but for the case of ε(q 2 = 0), which are shown as a comparison.

"bBG cuts").
At BESIII, the photon energy resolution of the EMC σ E /E = 2.3%/ E/GeV ⊕ 1% [36], and we take σ E = 40 MeV for all energy conservatively. At the BESIII, photon reconstruction efficiencies are all more than 99% [55], we assume them to be 100% in our paper. For the EMC at STCF, we assume the same energy resolution and reconstrunction efficiencies with BESIII to present a preliminary projection limit, because of the similarity of the two experiments. We take σ E = 25 (40, 50) MeV for √ s = 2, (4, 7) GeV. In addition to the "bBG cuts", we select final photon in the energy window of |E γ − (s − m 2 Z )/(2 √ s)| < σ E /2 (hereafter the "optimized cut") to enhance the discovery sensitivity.
At BESIII, since 2012 monophoton trigger has been implemented and the corresponding data luminosity reach about 14 fb −1 with the CM energy from 2.125 GeV to 4.6 GeV [56]. We define 1 case. The exisiting constraints are also presented in the shaded region, and the summary for these limits from different experiments can be found in Fig.3. The red band shows the region that could explain the muon anomalous magnetic moment (g − 2) µ ± 2σ. We present three expected limits with different experiments at Belle II, 1. γ + INV channel with bBG and gBG considered.
We translate the constraints on the dark photon from the search of invisible decay at Belle II assuming a 20 fb −1 dataset [35], where the bBG and gBG are all considered, to L µ − L τ gauge boson using the relation of Eq. 2. γ + INV channel with only bBG considered. We compute the limits without gBG taking into account as mentioned above. The "bBG cuts" are applied to remove the reducible BG events and only the irreducible BG contribute to the BG events if gBG is not considered. After the "optimized cut", we show the 95% C.L. upper bound on g Z at Belle II with the integrated luminosity of 50 ab −1 in Fig.9, which is lablled as "Belle II γ + INV" for the kinetic mixing, which is lablled as "Belle II µ + µ − + INV" One observes that on the searches for the invisible decay of Z , the sensitivity at 50 ab −1 Belle II with µ + µ − + INV channel is slightly better with the γ + INV channel. It can also be found that these two results are already excluded by current constraints. While without the gBG considered in the γ + INV channel, the sensitivity can be improved almost 1 order and the gauge coupling constant g Z down to about 4.2 × 10 −4 when m Z < 2m µ , which still left a thin slice of mass region ∼ (0.01 − 0.03) GeV to explain the moun (g − 2) anomaly. The one order of magnitude difference in sensitivity between the two Belle II limits via the monophoton search, shows that the control on gGB is very important in probing the Z parameter space.
The STCF and BESIII limits are obtained when the BG due to the gaps in the detectors are neglected, since BESIII did not released any analysis about gBG. We emphasize that more rigorous BESIII and STCF sensitivities could be obtained with such gBG anlysis available in the future. With about 14 fb −1 integrated luminosity collected during 2012-2018 [56] the upper limits from BE-SIII are exclued by CCFR experiment. The STCF limits are presented at √ s = 2, (4, 7) GeV with the integrated luminosity of 30 ab −1 . The future monophoton searches at the STCF experiment operated at √ s = 2−7 GeV can eliminate the moun g − 2 favored window when m Z 5 GeV. In the low mass region, 2 GeV STCF provide best sensitivity since the signal to BG ratio increases when the colliding energy decreases, and g Z can be down to about 4.2 × 10 −5 when m Z < 2m µ , which is improved about 1 order than the monophoton searches at 50 ab −1 Belle II with gBG omitted.
In Fig.10, we present the dependence for exclusion regions of g Z corresponding to m Z = 0.1 GeV on the mass ratio r = m L2 /m L1 and r = m S2 /m S1 via monophoton searches from BESIII with 14 fb −1 , Belle II with 50 ab −1 and future 4 GeV STCF with 30 ab −1 . The shaded grey region is already excluded by CCFR experiments, which is independent on the mass ratio. One can see that g Z can down to 1.3 (2.7) × 10 −5 when m L2 /m L1 (m S2 /m S1 ) = 100 at 4 GeV STCF with 30 ab −1 .

SUMMARY
In this paper, we probe the invisible decay of the L µ − L τ gauge boson via monophoton signature at three different electron colliders operated at the GeV scale: Belle II, BESIII, and STCF. In the minimal U (1) Lµ−Lτ model, we extend the SM with a U (1) Lµ−Lτ gauge symmetry and assume that the kinetic mixing term between Z and photon is absent at tree level, but can arise at one loop level due to µ and τ leptons. We also further extend the minimal U (1) Lµ−Lτ model with extra heavy vector-like leptons or charged scalars, where the additional contributions to the kinetic mixing arising from extra particles inside the loop. The exciting nondecoupling behavior of the contribution since the extra heavy vector-like leptons or charged scalars to the kinetic mixing is also demonstrated. The visible signatures of heavy leptons or charged scalars, too heavy to be directly de- : The sensitivity on gauge coupling g Z at Belle II, BESIII and STCF. Notice that we do not include the gBG analysis for BESIII and STCF limits. The solid lines indicate the case of that Z has no additional decay channel to dark matter, and the dotted lines indicate Br(Z → invisible) 1 cases. The exisiting constraints from different experiments are presented in the shaded region, and summarized in Fig.3. The red band shows the region that could explain the muon anomalous magnetic moment (g − 2)µ ± 2σ. The BESIII limit is obtained with the 14 fb tected at high energy colliders, maybe possible in processes modified by the γ − Z mixing.
We translate the sensitivity for dark photon within monophoton signature projected by Belle II to U (1) Lµ−Lτ gauge boson taking into account various SM BGs. We also recast the recent invisible search of Z in the µ + µ − Z production at Belle II. It is found that, By ignoring the BG due to the gaps in the detectors, we present the constraints at BESIII with 14 fb −1 luminosity and at future 30 ab −1 STCF. For comparison, we also compute the limits at 50 ab −1 Belle II without gBG taking into account. It is found that the future 2 GeV STCF can further improve the sensitivity to low mass Z than Belle II via monophoton signature since it is operated at lower energy. The future STCF can exclude the moun g − 2 anomaly favored parameter region when m Z 5 GeV. And gauge coupling constant g Z in the minimal U (1) Lµ−Lτ model can be probed down to about 4.2 × 10 −5 when m Z < 2m µ at future 30 ab −1 STCF with √ s = 2 GeV. The shaded grey region is already excluded by CCFR experiments, which is independent on the mass ratio.