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Searching for the light leptophilic gauge boson Zx via four-lepton final states at the CEPC

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Chong-Xing Yue, Yan-Yu Li, Mei-Shu-Yu Wang and Xin-Meng Zhang. Searching for the light leptophilic gauge boson Zx via four-lepton final states at the CEPC[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad25f5
Chong-Xing Yue, Yan-Yu Li, Mei-Shu-Yu Wang and Xin-Meng Zhang. Searching for the light leptophilic gauge boson Zx via four-lepton final states at the CEPC[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad25f5 shu
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Searching for the light leptophilic gauge boson Zx via four-lepton final states at the CEPC

  • 1. Department of Physics, Liaoning Normal University, Dalian 116029, China
  • 2. Center for Theoretical and Experimental High Energy Physics, Liaoning Normal University, Dalian 116029, China

Abstract: We investigate the possibility of detecting the leptophilic gauge boson Zx predicted by the U(1)LeLμ model via the processes e+e+Zx(Zx+) and e+e+Zx(Zxν¯ν) at the Circular Electron Positron Collider (CEPC) with a center of mass energy s=240 GeV and luminosity L=5.6ab1. We provide the expected sensitivities of the CEPC to the parameter space at the 1σ, 2σ, 3σ, and 5σ levels.

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    I.   INTRODUCTION
    • The Circular Electron Positron Collider (CEPC) [1] is a particle physics research program of great scientific significance and has great potential. The concept of the CEPC is developed in the context of many international large colliders, such as the Large Hadron Collider (LHC) at the European Center for Nuclear Research (CERN). In contrast to previous colliders with high energy consumption and costs as well as the pressure of data processing and storage, the CEPC has unique features and advantages. Firstly, the CEPC is an electron collider in which positrons and electrons collide with each other to produce high-energy particle events. Unlike hadron collisions, electron collisions produce particle events that are much clearer and more controllable, facilitating precise measurements and particle identification. Secondly, the CEPC plans to build a highly detailed detector that will be able to capture and record all the important information in particle collisions, providing physicists with a large amount of data to study the behavior of elementary particles. In addition, the CEPC will invest significant effort in improving data processing and storage techniques to cope with the high density of collision data. Finally, the CEPC has a much brighter and cleaner experimental environment. The standard model (SM) observables can be studied with unprecedented precision, and the precision of many electroweak observables can be improved by an order of magnitude or more. Therefore, the CEPC offers an unmatched opportunity for precision measurements and searches far beyond the standard model (BSM) physics.

      Among the many new physics (NP) scenarios, a class of models can predict the existence of leptophilic gauge boson Zx; this kind of new neutral gauge boson arises due to the extension of a group in the standard model (SM) with the U(1)LxLy for x,y{e,μ,τ} [24]. The global symmetry U(1)LxLy can be introduced to the SM, which is anomaly-free, without any additional particles [5, 6]. When the U(1)LxLy gauge symmetry is spontaneously broken, the leptophilic gauge boson Zx gains mass. This class of models can be a good solution to solve some problems in the SM, such as the neutrino mass and mixing problem [79], the dark matter dark energy problem [912], and the muon anomalous magnetic moment problem [1315]. In our paper, we discuss the possibility of probing this class of the leptophilic gauge boson Zx at the CEPC.

      The study of the leptophilic gauge boson Zx is an important step in exploring NP. The Zx boson can be produced at current collider experiments. For example, at the LHC, the leptophilic gauge boson Zx is mainly produced via the Drell-Yan process, where the Zx is radiated from the final-state leptons; the constraints on the Zx boson can be given via the processes pp4,3+EmissT,2+EmissT, or 1+EmissT [1619]. At the KEKB collider, the leptophilic gauge boson Zx is produced via the process e+eμ+μZx(Zxμ+μ) in the framework of the U(1)LμLτ model in the small mass range MZx<10 GeV [20]. The processes e+eZx+ or qˉq [21] and e+eγZx(+,μ+μ) [22] can also be used to search for the Zx boson at the LEP and BABAR. Most of the LHC (and Tevatron) bounds coming from resonance searches do not directly apply to such a neutral leptophilic sector. The relevant collider constraints of the U(1)LeLμ model mainly come from the LEP and are generally much weaker than the direct LHC constraints applicable for hadrophilic resonances [23]. Therefore, the futuree+e colliders are uniquely capable of probing the leptophilic gauge boson Zx to unprecedented mass and coupling values. Previous studies [24, 25] have investigated the sensitivity of the process e+eZxγ to explore the leptophilic gauge boson Zx in future e+e colliders. In general, properties of any new particle can be studied via different processes, even for the same collider experiments. Furthermore, we find that there are few studies that search for the gauge boson Zx predicted by the U(1)LeLμ model via four-lepton final state processes at future e+e colliders; hence, we propose searching for this kind of leptophilic gauge boson Zx via the processes e+e+Zx(Zx+) and e+e+Zx(Zxν¯ν) at the 240 GeV CEPC. We expect these processes to give better sensitivities in the certain mass range.

      The remainder of the paper is organized as follows. Section II briefly introduces the U(1)LeLμ model and summarizes the constraints of existing experiments on the model. Based on the details of the analysis of the Zx signal processes e+e+Zx(Zx+) and e+e+Zx(Zxν¯ν) and the relevant SM backgrounds, the sensitivity projections of the CEPC to the U(1)LeLμ model parameter space are presented and compared with other experimental results in Section III. Finally, the conclusion and discussion are given in Section IV.

    II.   THE U(1)LeLμ MODEL
    • The U(1)LxLy model [24] comprises the SM gauge group SU(3)CSU(2)LU(1)Y expanding a U(1)LxLy group without introducing an anomaly. This surprising feature is the main motivation considered here. For convenience, in Table 1, we list the lepton charges for the U(1)LxLy models. Here, e, μ, and τ are three generations of charged leptons; νe, νμ, and ντ represent the corresponding left-handed neutrinos, respectively.

      Model Charge
      e,νe μ,νμτ,ντ
      LeLμ 1 10
      LeLτ 1 01
      LμLτ 0 Le12(Lμ+Lτ)1
      Le12(Lμ+Lτ) 1 1212
      Le+2(Lμ+Lτ) 1 22

      Table 1.  Lepton charges corresponding to the U(1)LxLy models.

      The Lagrangian part of the U(1)LeLμ model can be written as

      L(Zx)=gZxα[Qe(ˉeγαe+¯νeγαPLνe)+Qμ(ˉμγαμ+¯νμγαPLνμ)+Qτ(ˉτγατ+¯ντγαPLντ)]14ZxμνZxμν+12m2ZxZxμZxμ,

      (1)

      where the gauge coupling constant is denoted as g; PL=12(1γ5) is the left chirality projector; and Qe, Qμ, and Qτ correspond to the charges of leptons of three generations in the U(1)LeLμ model. The Zx field strength tensor can be written as

      Zxμν=μZxννZxμ.

      (2)

      Before we discuss the experimental constraints on the gauge boson Zx, we present the decays of the Zx boson. The partial decay width of Zx+(ν¯ν) for a single flavor lepton is given by

      Γ(Zx+)=(gQ)2MZx12π(1+2m2M2Zx)14m2M2Zx,

      (3)

      Γ(Zxν¯ν)=(gQ)2MZx24π.

      (4)

      In the U(1)LeLμ model, the gauge boson Zx can only couple to two flavor leptons; therefore, the decay channels of the Zx boson are as follows:

      Zxe+e, Zxμ+μ, Zxνe¯νe, Zxνμ¯νμ.

      (5)

      Since MZxM, we can neglect the mass of the lepton in Eq. (3), which gives the total width of the gauge boson Zx as

      ΓZxg24πMZx.

      (6)

      There are two possible ways to discover the Zx boson. On the one hand, the Zx boson is heavy at the current energy, and we would need a higher energy to find it. On the other hand, it may be that the Zx mass is very small, and the coupling to the particles in the SM is weak (similar to the search for the Higgs boson); hence, researchers search for it by directly or indirectly producing it at future colliders. In our work, we prefer the latter .When the boson Zx has a small mass, the τ mass is heavy and unstable, and we mainly consider that the Zx boson couples only to the e and μ subsets and their corresponding neutrinos in the U(1)LeLμ model. Some existing constraints on the leptophilic gauge boson mass MZx and coupling g in the U(1)LeLμ model are summarized in Ref. [26]. The LEP bounds give the most stringent bounds in the larger mass range MZx103 GeV at 1σ(2σ) via e+e+ processes. CMS investigated the final state 4μ for the case that all muons originate from the decay of an (almost) on-shell Z boson, offering good sensitivity for 10 GeV <MZx<60 GeV. The strongest constraints on the coupling g with the 1060 GeV mass range come from the LHC at a 95% confidence level (CL). The g can be as low as 2×102 [19, 23]. The production of a muon-antimuon pair in the scattering of muon neutrinos in the Coulomb field of a target nucleus gives a strong bound, e.g., neutrino trident production [27, 28]. A combination of measurements of the trident cross section from CHARM-II [29], CCFR [30], and NuTeV [31] imposes a bound of g1.9×103MZx/GeV on the U(1)LeLμ model [23]. The sensitivities from (g2)e and (g2)μ on g in the U(1)LeLμ model are also considered; their results are in the range 0.21 and 4×1021, respectively [23, 3240]. Hence, we propose the process e+e+Zx(Zx+ or ν¯ν) in the U(1)LeLμ model with 10 GeV MZx60 GeV to progress further in the search for expected sensitivities of the Zx boson at thes=240 GeV CEPC.

    III.   SEARCHING FOR Zx AT THE CEPC
    • The main Feynman diagrams of the signal process e+e+Zx(Zx+ or ν¯ν) are shown in Fig. 1, which can be expanded into the following four processes: e+ee+eZx(Zxe+e)e+ee+e, e+eμ+μZx(Zxμ+μ)μ+μμ+μ, e+ee+eZx(Zxνe¯νe)e+eνe¯νe, and e+eμ+μZx(Zxνμ¯νμ)μ+μνμ¯νμ. In Fig. 2, we present the cross sections of four signaling processes and the corresponding backgrounds. The numerical results for the cross sections are imposed on the basic cuts. We make the transverse momenta of the leptons PT() greater than 10 GeV, and the absolute value of the lepton pseudorapidity η needs to be less than 2.5. These basic cuts are then summed up as

      Figure 1.  Main Feynamn diagrams for the process e+e+Zx(Zx+ or ν¯ν) within {e,μ}.

      Figure 2.  (color online) Cross sections of the signal and background processes as functions of the mass Zx when the coupling limits g=0.01GeV1.

      PT()>10GeV,η∣<2.5.

      (7)

      When the leptophilic gauge boson Zx decays to a pair of neutrinos, the beam polarizations can help further suppress the SM backgrounds to enhance the signals [41]. Therefore, in the right panel of Fig. 2, we show the polarized cross sections of the e+e+ν¯ν processes with the beam polarization configurations (Pe+,Pe)=(30%,+80%). The solid lines represent the signal cross sections, and the dashed lines represent the SM background cross sections. The cross section ranges of signal processes e+ee+eνe¯νe and e+eμ+μνμ¯νμ for 10 GeV MZx60 GeV are 3.24×1041.6×103 pb and 1.7×1042.24×104 pb , respectively, with g=0.01GeV1. For the above two processes, the background cross sections are 0.04823 pb and 0.03663 pb, respectively.

      The left panel shows the Zx boson decaying to a pair of leptons. We consider the effect of polarization on the process Zx+, but the variations in the cross sections are not significant; hence, we do not impose beam polarizations on the cross sections. The solid-red and solid-black lines represent the cross sections of the signal processes e+ee+ee+e and e+eμ+μμ+μ, respectively. The numerical results are 2.31×1044.24×104 pb and 2.49×1058.91×105 pb, respectively, in the mass range 10 GeV MZx60 GeV when the coupling constant g=0.01GeV1. The dashed-red and dashed-black lines represent the cross sections of the background processes e+ee+ee+e(0.01477 pb) and e+eμ+μμ+μ (0.001899 pb), respectively. The cross sections of signals in the parameter region are smaller than the cross sections of corresponding SM backgrounds.

      Next, we used FeynRules [42] to simulate the signals to produce a model file output in the UFO format. Then, all signal and background events were simulated using MadGraph5 [43]; the parton shower and hadronization were carried out with Pythia8 [44], while the detector simulation was performed using MadAnalysis5 [45] and Delphes3 [46]. In our analysis, we generate, in each case, 10k signal events in an interval where the mass of Zx increases in order from 10 GeV to 60 GeV, with 500k events for backgrounds.

    • A.   Visible decay channel Zx+

    • To further improve event selection, the signal and background distributions of the angular separation R between two muons, which is defined as R=(ϕ)2+(η)2, and invariant masses M(μ+,μ) are shown in Fig. 3. We can see that the background and signal have very distinctive characteristics. In particular, for the distribution of invariant masses M(μ+,μ), the peaks in M(μ+,μ) still denounce the presence of signals, making the distinction against the smooth background easy. We select M(μ+,μ)MZx5. R is greater than 0.5 for Zx mass from 10 GeV to 30 GeV and greater than 0.7 when the Zx mass is in the mass range 3060 GeV for the process e+eμ+μμ+μ. Based on the characteristics of the kinematic distributions, the selected cuts are listed in Table 2. After these improved cuts are applied, the SM background is significantly depressed. We take a signal benchmark point every 10 GeV in the 1060 GeV mass interval and display the cross sections of the signal and background after applying the above selection cuts for these benchmark points for g=0.01GeV1 at the 240 GeV CEPC with L=5.6ab1 in Table 3. We also show the statistical significance (SS) in the last column of Table 3, which is defined as SS=S/S+B, where S represents the number of signal events, and B represents the number of background events. The 1σ, 2σ, 3σ, and 5σ regions in the g-MZx plane are plotted in Fig. 4. The expected bounds on g can reach 6.2×103 (8.1×103) GeV1 at 3σ(5σ) levels. For the same signal process for the mass MZx<10 GeV, Ref. [20] provides the upper limit on g at SS =1σ level; however, we can provide the SS at 3σ(5σ) levels for the mass range 1060 GeV. Thus, the CEPC has the potential to discover the Zx boson in the considered mass range.

      Figure 3.  (color online) Normalized distributions of R (a) and M(μ+,μ) (b) from the signal and background events for different MZx benchmark points for the process e+eμ+μμ+μ at the CEPC with s=240 GeV and L=5.6ab1.

      Cut Mass
      10GeVMZx30GeV 30GeV<MZx60GeV
      Cut1 R>0.5 R>0.7
      Cut2 M(μ+,μ)MZx5 M(μ+,μ)MZx5

      Table 2.  Improved cuts for the process e+eμ+μμ+μ.

      Cross sections for signal (background)/fb
      MZx/GeVBasic cuts Cut1 Cut2 SS
      102.4852×102 2.4804×102 2.3443×102 4.7640
      (1.899)(1.894)(0.112)
      206.7516×102 6.7369×102 6.3763×102 7.4520
      (1.899)(1.894)(0.346)
      308.2532×102 8.2393×102 7.8353×102 7.1090
      (1.899)(1.894)(0.601)
      408.7872×102 8.7711×102 8.3870×102 6.4464
      (1.899)(1.894)(0.864)
      508.8983×102 8.8865×102 8.5551×102 5.8548
      (1.899)(1.894)(1.109)
      608.8160×102 8.8096×102 8.5297×102 5.3669
      (1.899)(1.894)(1.325)

      Table 3.  Cross sections of the signal and background after imposing the improved cuts for g=0.01GeV1 at the CEPC with s=240 GeV and L=5.6ab1 for the process e+eμ+μμ+μ.

      Figure 4.  (color online) 1σ, 2σ, 3σ, and 5σ regions for the process e+eμ+μμ+μ at the CEPC with s=240 GeV and L=5.6ab1 in the g-MZx plane.

      For the case where the gauge boson Zx decays into a pair of electrons, the kinematic distributions of the signal process e+ee+ee+e, R, ηe, ηe+, and M(e+,e) are shown in Fig. 5. The mass of Zx is greater than 40 GeV, and the distributions of the peaks of ηe and ηe+ are shifted. Therefore, we divide the mass range into two segments of 1040 GeV and 4060 GeV when we select the effective cuts. Ultimately, we summarize the specific cuts in Table 4. After applying improved cuts, the cross sections of the signal and the background are shown in Table 5. We also present the regions of SS at 1σ, 2σ, 3σ, and 5σ levels in Fig. 6. As can be seen from the figure, the sensitivity projections of Zx become weaker with increasing mass, and there is a significant dip at MZx=30 GeV with ​g=5×103GeV1. By comparing the above two processes, the four-electron final state is more sensitive to discovering the Zx boson.

      Figure 5.  (color online) Normalized distributions of R (a), ηe (b), ηe+ (c), and M(e+,e) (d) from the signal and background events for different MZx benchmark points for the process e+ee+ee+e at the CEPC with s=240 GeV and L=5.6ab1.

      Cut Mass
      10GeVMZx40GeV 40GeV<MZx60GeV
      Cut1 R>0.7 R>1
      Cut2 ηe>1.4 ηe>1.1
      Cut3 ηe+<1.4 ηe+<1.1
      Cut4 M(e+,e)MZx5 M(e+,e)MZx5

      Table 4.  Improved cuts for the process e+ee+ee+e.

      Cross sections for signal (background)/fb
      MZx/GeVBasic cuts Cut1 Cut2 Cut3 Cut4 SS
      100.2317 0.2178 0.1965 0.1784 0.1533 10.7950
      (14.77)(14.18)(12.30)(10.77)(0.9745)
      200.3526 0.3360 0.3124 0.2905 0.2465 10.9270
      (14.77)(14.18)(12.30)(10.77)(2.602)
      300.4241 0.4061 0.3765 0.3495 0.3027 10.2330
      (14.77)(14.18)(12.30)(10.77)(4.596)
      400.3876 0.3698 0.3323 0.2984 0.2674 8.3090
      (14.77)(14.11)(11.27)(9.133)(5.532)
      500.3261 0.3108 0.2839 0.2588 0.2357 6.6985
      (14.77)(14.11)(11.27)(9.133)(6.696)
      600.2672 0.2544 0.2338 0.2148 0.1984 5.3463
      (14.77)(14.11)(11.27)(9.133)(7.508)

      Table 5.  Cross sections of the signal and background after imposing the improved cuts for g=0.01GeV1 at the CEPC with s=240 GeV and L=5.6ab1 for the process e+ee+ee+e.

      Figure 6.  (color online) 1σ, 2σ, 3σ, and 5σ regions for the process e+ee+ee+e at the CEPC with s=240 GeV and L=5.6ab1 in the g-MZx plane.

    • B.   Visible decay channel Zxν¯ν

    • If the Zx boson decays to a pair of neutrinos, the processes e+ee+eνe¯νe and e+eμ+μνμ¯νμ have larger cross sections compared to those for the case when the Zx boson decays to a pair of leptons. For the process e+eμ+μνμ¯νμ, according to the kinetic distributions in Fig. 7, the transverse momentum PT(), invariant mass M(μ+,μ), angular separation R between two muons, and transverse energy ET are improved cuts, as presented in Table 6 for the entire mass range MZx=1060 GeV. Optimized cuts might preserve as many signal events as possible. Then, we provide the signal and background cross sections after imposing the optimized cuts for the process in Table 7. We can see that, when the background is suppressed by two orders of magnitude, the signal is substantially preserved. Figure 8 shows the SS = 1σ,2σ,3σ,5σ ranges in the g-MZx plane; the constraints on the Zx boson are very strict with the coupling constant g reaching 6.7×103GeV1 at SS =5σ.

      Figure 7.  (color online) Normalized distributions of PT() (a), M(e+,e) (b), R (c), and ET (d) from the signal and background events for different MZx benchmark points for the process e+eμ+μνμ¯νμ at the CEPC with s=240 GeV and L=5.6ab1.

      Cut Mass
      10GeVMZx60GeV
      Cut1 PT()>5
      Cut2 M(μ+,μ)MZx∣≤5
      Cut3 R<4
      Cut4 ET<150

      Table 6.  Improved cuts for the process e+eμ+μνμ¯νμ.

      Cross sections for signal (background)/fb
      MZx/GeVBasic cuts Cut1 Cut2 Cut3 Cut4 SS
      100.1708 0.1706 0.03665 0.03665 0.03665 5.4300
      (36.63)(36.59)(0.2205)(0.2205)(0.2200)
      200.2122 0.2215 0.1054 0.1054 0.1054 10.1420
      (36.63)(36.59)(0.4993)(0.4993)(0.4985)
      300.2216 0.2215 0.1317 0.1317 0.1317 11.0030
      (36.63)(36.59)(0.6723)(0.6721)(0.6711)
      400.2229 0.2215 0.1317 0.1317 0.1414 10.8980
      (36.63)(36.59)(0.8064)(0.8056)(0.8046)
      500.2205 0.2204 0.1467 0.1467 0.1467 10.5890
      (36.63)(36.59)(0.9205)(0.9195)(0.9195)
      600.2153 0.2153 0.1443 0.1443 0.1443 9.9480
      (36.63)(36.59)(1.037)(1.036)(1.034)

      Table 7.  Cross sections of the signal and background after imposing the improved cuts for g=0.01GeV1 at the CEPC with s=240 GeV and L=5.6ab1 for the process e+eμ+μνμ¯νμ.

      Figure 8.  (color online) 1σ, 2σ, 3σ and 5σ regions for the process e+eμ+μνμ¯νμ at the CEPC with s=240 GeV and L=5.6ab1 in the g-MZx plane.

      When Zx decays to νe¯νe, the peak distribution of the e+ energy for the signal process e+ee+eνe¯νe is clearly demarcated from the background in Fig. 9. For the low mass range MZx=1040 GeV, the e+ energy retains more signals after applying the cuts. On the contrary, for the large mass range MZx=4060 GeV, the signal events of the invariant mass M(e+,e) outnumber the signal events of E(e+) after improving cuts; therefore, we add to the effective cuts at MZx=40 GeV, as shown in Table 8. Finally, Table 9 presents the cross sections of the signal and background after the improved cuts are imposed on the e+ee+eνe¯νe process, and we plot 1σ, 2σ, 3σ and 5σ ranges in Fig. 10. The sensitivity projections of the Zx boson that we obtain are very strict for the process, especially in the region of mass MZx=1040 GeV. In contrast to the three processes mentioned above, this process is more sensitive to the Zx boson.

      Figure 9.  (color online) Normalized distributions of PT() (a), E(e+) (b), and M(e+,e) (c) from the signal and background events for different MZx benchmark points for the process e+ee+eνe¯νe at the CEPC with s=240 GeV and L=5.6ab1.

      Cut Mass
      10GeVMZx40GeV 40GeV<MZx60GeV
      Cut1 PT()>5 PT()>5
      Cut2 E(e+)>110 M(e+,e)MZx∣≤5

      Table 8.  The improved cuts for the process e+ee+eνe¯νe.

      Cross sections for signal (background)/fb
      MZx/GeVBasic cuts Cut1 Cut2 SS
      101.6011 1.5658 0.49924 51.6830
      (48.23)(46.75)(0.02334)
      200.8491 0.8312 0.1709 28.9830
      (48.23)(46.75)(0.02334)
      300.5879 0.5761 0.06803 16.7940
      (48.23)(46.75)(0.02334)
      400.4583 0.4491 0.02810 9.2220
      (48.23)(46.75)(0.02334)
      500.3755 0.3677 0.1192 7.3630
      (48.233)(46.75)(1.522)
      600.3247 0.3172 0.1187 6.6180
      (48.23)(46.75)(0.1187)

      Table 9.  Cross sections of the signal and background after imposing the improved cuts for g=0.01GeV1 at the CEPC with s=240 GeV and L=5.6ab1 for the process e+ee+eνe¯νe.

      Figure 10.  (color online) 1σ, 2σ, 3σ, and 5σ regions for the process e+ee+eνe¯νe at the CEPC with s=240 GeV and L=5.6ab1 in the g-MZx plane.

    IV.   CONCLUSION AND DISCUSSION
    • Recently, there have been many studies related to the leptophilic gauge boson Zx. The search for the mass range of 10500 GeV Zx in the U(1)LμLτ model is widely studied at the LHC. However, the search for a small mass of Zx is very limited in the U(1)LeLμ model at the future e+e colliders. It is evident that there is still a large parameter space around the electroweak scale for us to explore the Zx boson [23]. Thus, we can search for theZx predicted by the U(1)LeLμ model at the CEPC to facilitate the extension of the sensitivity of Zx or stricter couplings. In our study, we investigate the prospects of the CEPC to unravel the NP associated with a new weak interaction, and the gauge bosonZx only couples to the e and μ subsets in the U(1)LeLμ model.

      We investigated the sensitivity of the CEPC with s=240 GeV and L=5.6ab1 to the coupling parameter g for MZx=1060 GeV. As can be seen from the four processes explored in the previous sections, the expected bounds of the process e+ee+eνe¯νe on g can reach 1.0×103 (1.6×103) GeV1 for MZx=1040 GeV at 3σ (5σ); these are the strictest constraints on the U(1)LeLμmodel. However, in the Zx mass range of 4060 GeV, the strictest constraints come from the process e+eμ+μνμ¯νμ, and the expected bounds on g can reach 5.1×103 (6.7×103) GeV1 at 3σ (5σ). Compared to the other three processes, the process e+eμ+μμ+μ is less constrained.

      In conclusion, the expected sensitivities of the four processes to the parameter space of the U(1)LeLμ models are different. However, when we compare our numerical results in Fig. 2 with those in Ref. [23], they are not experimentally excluded except from the process e+eμ+μμ+μ. At the same time, Ref. [24] indicates that the sensitivity to g for the process e+eZxγ can be as low as 5×103 in the mass range of 1060 GeV at the 2σ level. Our results can reach 1×103 for MZx=10 GeV via the process e+ee+eνe¯νe, and the constraints from the process e+eμ+μνμ¯νμ can reach 4.2×103 at the 2σ level in the entire mass range of 1060 GeV. Our numerical results align with those of Ref. [24], as the same conclusions are applicable to the process e+ee+ee+e. Hence, searching for the Zx boson predicted by the U(1)LeLμ model at the 240 GeV CEPC via processes e+e+Zx(Zxν¯ν or ) can enhance the sensitivity projections to the parameter space and promote further exploration of future e+e colliders for the U(1)LeLμ model, providing more opportunities for further discoveries regarding the leptophilic gauge boson Zx.

    ACKNOWLEDGEMENTS
    • Yan-Yu Li would like to thank Han Wang for very useful discussions.

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