Search for invisible decays of the Higgs boson produced at the CEPC

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Yuhang Tan, Xin Shi, Ryuta Kiuchi, Manqi Ruan, Maoqiang Jing, Dan Yu, Kaili Zhang, Xinchou Lou, Xin Mo, Gang Li and Susmita Jyotishmati. Search for invisible decays of the Higgs boson produced at the CEPC[J]. Chinese Physics C.
Yuhang Tan, Xin Shi, Ryuta Kiuchi, Manqi Ruan, Maoqiang Jing, Dan Yu, Kaili Zhang, Xinchou Lou, Xin Mo, Gang Li and Susmita Jyotishmati. Search for invisible decays of the Higgs boson produced at the CEPC[J]. Chinese Physics C. shu
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Search for invisible decays of the Higgs boson produced at the CEPC

  • 1. Institute of High Energy Physics, Beijing 100049, China
  • 2. School of Physical Sciences, University of Chinese Academy of Science (UCAS), Beijing 100049, China
  • 3. State Key Laboratory of Particle Detection and Electronics, 19B Yuquan Road, Shijingshan District, Beijing 100049, China
  • 4. Department of Physics, University of Texas at Dallas, Texas 75080-3021, USA

Abstract: The Circular Electron Positron Collider (CEPC) proposed as a future Higgs boson factory will operate at a center-of-mass energy of 240 GeV and accumulate 5.6 ab-1 of integrated luminosity in 7 years. In this paper, we estimate the upper limit of BR($H \rightarrow$ inv) for three independent channels including two leptonic channels and one hadronic channel at the CEPC. Based on the full simulation analysis, the upper limit of BR($H \rightarrow$ inv) could reach 0.26% at the 95% confidence level. In the Stand Model (SM), the Higgs boson can only decay invisibly via $H\rightarrow ZZ^\ast\rightarrow\nu\overline{\nu}\nu\overline{\nu}$, so any evidence of invisible Higgs decays that exceeds BR($H \rightarrow$ inv) of SM will indicate a phenomenon that is beyond the SM (BSM). The invariant mass resolution of visible hadronic decay system $ZH(Z \rightarrow qq$, $ H \rightarrow$ inv) is simulated and the physics requirement at the CEPC detector to reach this is given.

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    1.   Introduction
    • Many pieces of cosmological evidence point towards the existence of dark matter (DM), such as rotation curves in galaxies, the masses of clusters of galaxies, and gravitational lensing of galaxies [1, 2]. However, there is no candidate for DM in the Stand Model (SM). In collider physics, the Higgs boson might be the portal between the new physics, such as DM and the fourth generation neutrino, and the SM sector [3-5]. In this case, the DM particles, which interact weakly with ordinary matter and are completely invisible in the detector, can be observed indirectly by studying the Higgs decays. In SM, the Higgs boson can only decay invisibly via $ H\rightarrow ZZ^\ast \rightarrow \nu\overline{\nu}\nu\overline{\nu} $, as shown in Fig. 1 and its branching ratio (BR) is 1.06$ \times10^{-3} $ [6]. Therefore, any evidence of invisible Higgs decays that exceeds this BR will indicate a phenomenon that is beyond the SM (BSM).

      Figure 1.  Feynman diagrams of Higgs boson invisible decays at an electron-positron collider such as the CEPC. In the process $e^+e^- \rightarrow ZH$, the invisible decays of Higgs boson are via $H\rightarrow ZZ^\ast \rightarrow \nu\overline{\nu}\nu\overline{\nu}$.

      The search for the invisible decays of the Higgs boson has been performed at the Large Hadron Collider (LHC). The signature for the invisible Higgs decays at the LHC is a large missing transverse momentum recoiling against a visible system. ATLAS and CMS respectively give a 95% confidence level (CL) upper limit of 26% [7] and 19% [8] on the Higgs boson invisible branching ratio (BR($ H \rightarrow $ inv)). They performed with ATLAS detector and CMS detector respectively using combined 4.7 fb−1, 20.3 fb−1, 36.1 fb−1 and combined 4.9 fb−1, 19.7 fb−1, 38.2 fb−1 of proton-proton collisions at the center-of-mass energy of 7 TeV, 8 TeV and 13 TeV at the LHC. Compared with the results from the LHC, Higgs boson candidates can be identified through a technique known as the recoil mass method without using its decays at the Circular Electron Positron Collider (CEPC) [6]. As an example of recoil mass method: in $ e^{+}e^{-} $$ \rightarrow $ $ ZH(Z \rightarrow f\bar{f} $, $ H \rightarrow $ inv) channel, the fermions (f) can be identified and their momentum can be measured. By selecting the fermion pair from the Z boson decay, the mass of the system recoiling against the fermion pair commonly known as the recoil mass $ M_{\rm{recoil}} $, which can be calculated with:

      $ M_{\rm{recoil}} = \sqrt{(\sqrt{s} - E_{f\bar{f}})^{2} - P_{f\bar{f}}^{2}} \ , $

      (1)

      where $ E_{f\bar{f}} $ and $ P_{f\bar{f}} $ are the total energy and momentum of two fermions respectively and $ \sqrt{s} $ is the center-of-mass energy. The $ M_{\rm{recoil}} $ distribution should show a peak at the Higgs boson mass around 125 GeV for $ f\bar{f} $$ \rightarrow ZH $ process. In this way, the properties of the Higgs boson can be measured precisely without reconstructing the Higgs boson by its decaying products. Therefore, the Higgs boson production can be disentangled from its decay in a model-independent way. Moreover, the $ e^+e^- $ collisions have a much reduced hadronic background (Higgs boson channels are signal) contamination compared to the hadron collisions, which allows better exclusive measurements of Higgs boson decay channels. The electron-positron Higgs factory is an essential machine for understanding the nature of the Higgs boson.

      CEPC is a Higgs factory proposed by the Chinese high energy physics community. CEPC is designed to deliver a combined integrated luminosity of 5.6 ab-1 to two detectors in 7 years. CEPC will operate at a center-of-mass energy $ \sqrt{s} $ $ \sim $ 240-250 GeV, and over one million Higgs boson events will be produced during this period. Thanks to the large statistics, the good beam energy spread is about 0.16% [9] and a new particle flow algorithm [10], the mass and width of the Higgs boson can achieve high precision measurement. With the SM $ ZH $ production rate, the upper limit of BR($ H \rightarrow $ inv) could reach 0.26% at the 95% CL, two orders of magnitude improvements over the ATLAS and CMS are expected.

      In the previous publication of CEPC, the upper limit of BR($ H \rightarrow $ inv) is 0.41% [6]. The previous paper used the CEPC-v1 detector while the CEPC-v4 detector is used in this paper. The main change from CEPC-v1 to CEPC-v4 is the reduction of the solenoidal field from 3.5 Tesla to 3.0 Tesla and the changing of $ \sqrt{s} $ of collider from 250 GeV to 240 GeV. Moreover, the reconstruction algorithm is different in two papers. Therefore, this paper does not involve the comparison of the two results.

      The paper performs three independent analyses corresponding to $ \mu\mu H $, $ eeH $, and $ qqH $ channels to estimate the upper limit of BR($ H \rightarrow $ inv) measurement at the CEPC. This paper is organized as follows: Section 2 is a brief introduction of the CEPC detector and Monte Carlo simulation. Section 3 is the introduction of the event selection of three channels. How to get the result of the upper limit and the dependence of the Boson Mass Resolution (BMR) will be discussed in section 4. Section 5 is the conclusion.

    2.   Detector design and Monte Carlo simulation
    • One of the physics programs of the CEPC is the precision measurements of the Higgs boson properties, and the CEPC detector should reconstruct and identify all key physics objects including charged leptons, photons, jets, missing energy, and missing momentum.

      The CEPC-v4 [6] detector is designed by using the International Large Detector (ILD) [11, 12] as a reference. The detector of CEPC-v4 is simulated using MokkaC [13] and Geant4 [14]. It is comprised of a tracking system, a Time-Projection-Chamber tracker (TPC), a high granularity calorimeter system, a solenoid of 3 Tesla magnetic field, and a muon detector embedded in a magnetic field return yoke. The tracking system consists of silicon vertexing and tracking detectors. The calorimetry system consists of electromagnetic calorimeter (ECAL) and Iron-Scintillator for the hadronic calorimeter (HCAL).

      The analysis is performed on Monte Carlo (MC) samples simulated at the CEPC-v4 detector. The Higgs boson signal and SM backgrounds at a center-of-mass energy of 240 GeV, corresponding to an integrated luminosity of 5.6 ab-1, are generated with WHIZARD1.95 [15]. The generated events are then processed with MokkaC, and attempt to reconstruct every visible particle with ARBOR [10]. The cross sections of major SM processes of $ e^{+}e^{-} $ collisions as a function of center-of-mass energy $ \sqrt{s} $ are used in the simulation, including Higgs production as well as the major backgrounds, where the initial-state radiation (ISR) effect has been taken into account. The Higgs boson signal and backgrounds are processed with Geant4 based full detector simulation and reconstruction. Limited by computing resources, about 20% of the two fermions backgrounds used the full simulation.

      All samples are grouped into signal and backgrounds, and the backgrounds are classified according to their final states. For the signal, this paper mainly focuses on the process of $ e^{+}e^{-}\rightarrow ZH $, which will be called a “$ ZH $” process. Then Z bosons will decay into leptons or hadrons, and the Higgs will decay into two Z bosons, which will decay into four neutrinos eventually. For the backgrounds, the major SM backgrounds are divided into the 2-fermion processes and the 4-fermion processes according to the final states. The 2-fermion backgrounds are $ e^{+}e^{-} $$ \rightarrow $ $ f\bar{f} $ where f refers to all lepton and quark pair except $ t\overline{t} $. The 4-fermion backgrounds are divided into 6 types: “single_z”, “single_w”, “szorsw”, “zz”, “ww”, and “zzorww”, which are shown in Table 1. For the processes, whose four final states are a pair of electrons and two other fermions, or a pair of electron neutrinos and two other fermions, the processes will be named “single_z”. “single_w” is the processes including one electron, one electron neutrino, and two other fermions. If the final states include a pair of electrons and a pair of electron neutrinos simultaneously, these processes are named “szorsw”. In addition to the above mentioned backgrounds, the same four fermions in the final states can be combined into different two intermediate bosons. If the two intermediate bosons can be two Z bosons, the processes will be named “zz''. The ``ww” process is that the two intermediate bosons can be two W bosons. If the two intermediate bosons can be two Z bosons or two W bosons, the processes belong to “zzorww”.

      Type Four final states
      single_z two electrons,two other fermions or two electron neutrinos,two other fermions
      single_w one electron,one electron neutrino,two other fermions
      szorsw a pair of electrons and a pair of electron neutrinos
      Type Two intermediate bosons
      zz two Z bosons
      ww two W bosons
      zzorww two Z bosons or two W bosons

      Table 1.  Six types of 4-fermion backgrounds.

    3.   Event selection
    • The signal of this analysis consists of three different channels, namely $ ZH(Z \rightarrow qq, H \rightarrow $ inv), $ ZH(Z \rightarrow \mu\mu, H \rightarrow $ inv), and $ ZH(Z \rightarrow ee, H \rightarrow $ inv). Table 2 shows detailed information of the Higgs boson decay channels. The observed upper limit on BR($ H \rightarrow $ inv) at 95% CL at the CMS is 19%, and expect CEPC to have more accurate results. In the event selection part, this analysis uses the BR($ H \rightarrow $ inv) = 10%, and the event selection is based on the distribution of signal and background. Below is the detailed event selection for each channel.

      Process Cross sections /fb Expected
      $ffH$ 203.66 1140496
      $e^{+}e^{-} H$ 7.04 39424
      ${\mu^+}{\mu^-} H$ 6.77 37912
      $q\bar{q} H$ 136.81 766136

      Table 2.  Cross sections of the Higgs boson production at $\sqrt{s}$ = 240 GeV and number of events expected in 5.6 ab-1.

    • 3.1.   $ ZH(Z \rightarrow qq, H \rightarrow $ inv)

    • In $ ZH(Z \rightarrow qq, H \rightarrow $ inv) process, due to the presence of quarks, there will be many final states. The event selection uses the information of all visible particles, and the distributions of the signal and backgrounds are shown in Fig. 2. The comprehensive event selections are as follows: In $ e^+e^- \rightarrow ZH(Z \rightarrow qq, H \rightarrow $ inv) process, the mass of the system recoiling against all visible particles from Z boson is $ M_{\rm{recoil}}^{\rm{visible}} $, which can be calculated with Eq.1 by replacing $ E_{f\bar{f}} $, $ P_{f\bar{f}} $ with $ E_{\rm{visible}} $, $ P_{\rm{visible}} $. $ E_{\rm{visible}} $ and $ P_{\rm{visible}} $ is the total energy and momentum of all visible particles. The peak of the $ M_{\rm{recoil}}^{\rm{visible}} $ distribution is close to the Higgs boson mass. Considering the resolution of the detector, the $ M_{\rm{recoil}}^{\rm{visible}} $ is limited to (100,150) GeV as shown in Fig. 2(a). To suppress 2-fermion backgrounds, the transverse momentum of all visible particles is required to satisfy $ P_{\rm{T}}^{\rm{visible}}> $ 18 GeV as shown in Fig. 2(b), and the difference of the azimuthed angles of the two jets should be less than 175°. Two jets are reconstructed from Z boson decay particles. $ E_{\rm{visible}} $ is the energy of all visible particles which can be described as:

      Figure 2.  (color online) The distributions of $M_{\rm{recoil}}^{\rm{visible}}$, $P_{\rm{T}}^{\rm{visible}}$, $E_{\rm{visible}}$, and $M_{\rm{visible}}$ for signal and backgrounds in the $ZH(Z \rightarrow qq, H \rightarrow$ inv) process with BR($H \rightarrow$ inv) = 10% are plotted with all cuts except the ones shown already applied in the figure and its subsequent cuts (based on Table 3). The blue arrows are the range of cut.

      $ E_{\rm{visible}} = \frac{s+M_{\rm{visible}}^{2}-(M_{\rm{recoil}}^{\rm{visible}})^{2}}{2\sqrt{s}} , $

      (2)

      where the $ M_{\rm{recoil}}^{\rm{visible}} $ is around 125 GeV, $ \sqrt{s} = 240 $ GeV and the invariant mass of visible system ($ M_{\rm{visible}} $) is equal to the Z boson mass which is 91.2 GeV. $ M_{\rm{visible}} $ should be limited to (85,102) GeV as shown in Fig. 2(c). Taken the value of parameters into Eq.2, the $ E_{\rm{visible}} $ should be near 105 GeV as shown in Fig. 2(d). According to the equation: $ P_{\rm{visible}}^2 = E_{\rm{visible}}^2 - M_{\rm{visible}}^2 $, $ P_{\rm{visible}} $ should be near 52 GeV. Due to the presence of quarks, the final states may include many charged particles. It is necessary to limit that the number of charged particles ($ N_{\rm{charged}} $) with energy greater than 1 GeV larger than 5. To suppress the backgrounds from tau particles, a dedicated tau-finding algorithm TAURUS has been developed [16]. Since the $ ZH(Z \rightarrow qq, H\rightarrow $ inv) process may include tau particles, the mass of all tau particles ($ M_{\tau} $) should be less than 95 GeV to suppress the backgrounds containing tau and quarks.

      Table 3 shows the yields of signal ($ qqH $_inv) and its backgrounds of the cut chain. The value of the significance [17] is used to judge the effect of the cuts. After the event selection, the signal selection efficiency is 60.81%, and the total background rejection efficiency is 99.97%. The backgrounds, which contain neutrinos and two quarks, account for 95% of the total remaining backgrounds. These backgrounds compositions are similar to the signal channel and are hard to suppress further.

      Process $qqH$_inv 2f single_w single_z szorsw zz ww zzorww ZH_visible total_bkg Significance
      Total generated 76614 801152072 19517400 9072952 1397088 6389432 50826216 20440840 1140496 909936496 2.54
      100 GeV$<M_{\rm{recoil}}^{\rm{visible}}<$150 GeV 73800 47294924 1388875 822729 229217 507567 1752827 658204 97387 52751730 10.16
      18 GeV$<P_{\rm{T}}^{\rm{visible}}<$60 GeV 67115 9165311 1000762 269328 152273 282630 1294265 462029 79965 12706563 18.81
      90 GeV$<E_{\rm{visible}}<$117 GeV 63912 5748712 595697 223049 92958 231058 785392 272518 33705 7983089 22.59
      85 GeV$<M_{\rm{visible}}<$102 GeV 53786 605791 238191 148850 39280 135641 392277 113043 18284 1691357 41.14
      $\Delta\phi_{\rm{dijet}}<175^ \circ $ 51911 390077 230273 141494 38359 129135 379931 109735 17395 1436399 43.06
      30 GeV$<P_{\rm{visible}}<$58 GeV 48572 241510 148607 69457 24393 46807 226883 74781 13466 845904 52.32
      $N_{\rm{charged}}>5,E_{\rm{charged}}>1$GeV 47772 7986 18399 62990 6 43728 121365 4110 11699 270283 89.36
      $M_{\tau}<$95 GeV 46589 7111 11044 59815 1 41180 104784 3126 11111 238172 92.58
      Efficiency 60.81% 0.00% 0.06% 0.66% 0.00% 0.64% 0.21% 0.02% 0.97% 0.03%

      Table 3.  Yields for backgrounds and $ ZH(Z \rightarrow qq, H \rightarrow $ inv) signal at the CEPC, with $ \sqrt{s} $ = 240 GeV, BR($ H \rightarrow $ inv) = 10% and integrated luminosity of 5.6 ab-1.

    • 3.2.   $ ZH(Z \rightarrow \mu^+\mu^-, H \rightarrow $ inv) and $ ZH(Z \rightarrow e^+e^-, H \rightarrow $ inv)

    • The $ ZH(Z \rightarrow \mu^+\mu^-, H \rightarrow $ inv) process and $ ZH(Z \rightarrow e^+e^-, H \rightarrow $ inv) process are similar, and the two processes will be introduced together. Firstly, it is natural that only a pair of oppositely charged muons or electrons is required in the visible final states. By selecting two muons or two electrons, many related parameters can be used to suppress the backgrounds. The event selections are as follows: The recoil mass of two muons ($ M_{\rm{recoil}}^{\mu^{+}\mu^{-}} $) and the recoil mass of two electrons ($ M_{\rm{recoil}}^{e^{+}e^{-}} $) can be calculated using the Eq.1. The peak of $ M_{\rm{recoil}}^{\mu^{+}\mu^{-}} $ and $ M_{\rm{recoil}}^{e^{+}e^{-}} $ distribution should be around Higgs boson mass 125 GeV. Considering the resolution of muon and electron, and the distribution of signal and backgrounds as shown in Fig. 3 and Fig. 4, the recoil mass should satisfy 120 GeV$ <M_{\rm{recoil}}^{\mu^{+}\mu^{-}}< $150 GeV or 120 GeV$ <M_{\rm{recoil}}^{e^{+}e^{-}}< $170 GeV, and the invariant mass of two muons ($ M_{\mu^{+}\mu^{-}} $) or two electrons ($ M_{e^{+}e^{-}} $) are closer to Z boson mass. To suppress the 2-fermion backgrounds, the transverse momentum of the muon pair ($ P_{\rm{T}}^{\mu^+\mu^-} $) and the electron pair ($ P_{\rm{T}}^{e^{+}e^{-}} $) are required to be more than 12 GeV as shown in Fig. 3(c) and Fig. 4(c). The angle of two muons ($ \Delta\phi_{\mu^+\mu^-} $) less than 175° or two electrons ($ \Delta\phi_{e^+e^-} $) less than 176° is also required to suppress the 2-fermion backgrounds. The visible energy ($ E_{\rm{visible}} $), which is described in Eq.2, is mainly the energy of two muons ($ E_{\mu^{+}\mu^{-}} $) or two electrons ($ E_{e^{+}e^{-}} $) from Z boson decays, the value of $ E_{\rm{visible}} $ is approximately 105 GeV as shown in Fig. 3(d) and Fig. 4(d). Taken the approximate value of $ M_{\mu^{+}\mu^{-}} $ and $ E_{\mu^{+}\mu^{-}} $ into relativistic energy-momentum relation: $ M_{\mu^{+}\mu^{-}}^2 = E_{\mu^{+}\mu^{-}}^2-P_{\mu^{+}\mu^{-}}^2 $, the value of $ \frac{E_{\mu^{+}\mu^{-}}}{P_{\mu^{+}\mu^{-}}} $ is close to 2, similarly for $ \frac{E_{e^+e^-}}{P_{e^+e^-}} $.

      Figure 3.  (color online) The distributions of $M_{\rm{recoil}}^{\mu^{+}\mu^{-}}$, $P_{\rm{T}}^{\mu^+\mu^-}$, $E_{\rm{visible}}$, and $M_{\mu^+\mu^-}$ for signal and backgrounds in the $ZH(Z \rightarrow \mu^+\mu^-, H \rightarrow$ inv) process with BR($H \rightarrow$ inv) = 10% are plotted with all cuts except the ones shown already applied in the figure and its subsequent cuts (based on Table 4). The blue arrows are the range of cut.

      Figure 4.  (color online) The distributions of $M_{\rm{recoil}}^{e^+e^-}$, $P_{\rm{T}}^{e^+e^-}$, $E_{\rm{visible}}$, and $M_{e^+e^-}$ for signal and backgrounds in the $ZH(Z \rightarrow e^+e^-, H \rightarrow$ inv) process with BR($H \rightarrow$ inv) = 10% are plotted with all cuts except the ones shown already applied in the figure and its subsequent cuts (based on Table 5). The blue arrows are the range of cut.

      Table 4 is the yields for $ \mu\mu H $_inv signal and its backgrounds. The remaining backgrounds containing two muons and two neutrinos account for 61% of the total backgrounds. These backgrounds have similar topology as the signal which is hard to suppress further. The remaining backgrounds containing muon, tau, and two neutrinos account for 38% of the total backgrounds, the algorithm TAURUS has no effect on increasing the significance of $ \mu\mu H $_inv signal.

      Process $\mu\mu H$_inv 2f single_w single_z szorsw zz ww zzorww ZH total_bkg Significance
      Total generated 3791 801152072 19517400 9072952 1397088 6389432 50826216 20440840 1140496 909936496 0.13
      $N_{\mu^{+}}=1,N_{\mu^{-}}=1$ 3370 22737312 36123 723402 0 702045 1255617 1223596 59978 26738073 0.65
      120 GeV$<M_{\rm{recoil}}^{\mu^{+}\mu^{-}}<$150 GeV 3286 652655 24 100435 0 62463 250819 112143 5708 1184247 3.02
      85 GeV$<M_{\mu^{+}\mu^{-}}<$97 GeV 2791 381056 0 10739 0 20857 16720 24419 4493 458284 4.12
      12 GeV$<P_{\rm{T}}^{\mu^{+}\mu^{-}}$ 2705 92200 0 9483 0 18257 15906 21063 4331 161240 6.72
      $\Delta\phi_{\mu^{+}\mu^{-}}<175^ \circ $ 2598 72197 0 8894 0 17029 14769 20231 4142 137262 6.99
      102 GeV$<E_{\rm{visible}}<$107 GeV 2273 62 0 1456 0 484 4379 5435 10 11826 20.28
      $\frac{E_{\mu^{+}\mu^{-}}}{P_{\mu^{+}\mu^{-}}}<2.4$ 2243 27 0 1344 0 440 3502 4090 6 9409 22.29
      Efficiency 59.17% 0.00% 0.00% 0.01% 0.00% 0.01% 0.01% 0.02% 0.00% 0.00%

      Table 4.  Yields for backgrounds and $ ZH(Z \rightarrow \mu^+\mu^-, H \rightarrow $ inv) signal at the CEPC, with $ \sqrt{s} $ = 240 GeV, BR($ H \rightarrow $ inv) = 10% and integrated luminosity of 5.6 ab-1.

      Table 5 is the yields for $ eeH $_inv signal and its backgrounds at the CEPC. The cut Vertex$ _{\tau}< $0.0011, which is the position of decay vertex, changes the value of significance from 10.91 to 13.79. Since the signal channel does not contain tau, the Vertex$ _{\tau} $ of signal channel is much smaller than backgrounds from tau. The final states of the remaining backgrounds, which are composed of two electrons and two neutrinos, account for 70% of the total background. These backgrounds are the same as the final particles of the signal channel. The final particles of the remaining backgrounds containing tau, electron, and two neutrinos account for 23% of the total background, and the information of tau particles cannot further suppress these backgrounds. In conclusion, the tau-finding algorithm TAURUS can improve the significance of Higgs invisible decays to a certain extent, but it cannot completely suppress the backgrounds containing tau.

      Process $eeH$_inv 2f single_w single_z szorsw zz ww zzorww ZH total_bkg Significance
      Total generated 3942 801152072 19517400 9072952 1397088 6389432 50826216 20440840 1140496 909936496 0.13
      $N_{e^{+}}=1,N_{e^{-}}=1$ 3472 120476492 1286971 1945217 1161098 134637 796860 292694 117024 126210993 0.31
      120 GeV$<M_{\rm{recoil}}^{e^{+}e^{-}}<$170 GeV 3179 6469896 515411 226357 288321 4901 12836 34015 12443 7564180 1.16
      71 GeV$<M_{e^{+}e^{-}}<99$GeV 2617 2415241 98434 69803 105255 453 926 10853 9153 2710118 1.59
      12 GeV$<P_{\rm{T}}^{e^{+}e^{-}}<$55 GeV 2511 1351168 87124 45901 91180 352 788 9396 8774 1594683 1.99
      $\Delta\phi_{e^{+}e^{-}}<176^ \circ $ 2397 462573 81317 37220 87409 208 712 8613 8456 686508 2.89
      101 GeV$<E_{\rm{visible}}<$107 GeV 1614 6555 15198 2820 17583 12 54 1175 47 43444 7.70
      $1.8<\frac{E_{e^+e^-}}{P_{e^+e^-}}<2.4$ 1455 1423 6634 1127 7685 4 17 393 23 17306 10.91
      ${\rm{Vertex}}_{\tau}$$<$0.0011 1393 323 2436 926 5967 1 7 86 9 9755 13.79
      Efficiency 35.34% 0.00% 0.01% 0.01% 0.43% 0.00% 0.00% 0.00% 0.00% 0.00%

      Table 5.  Yields for backgrounds and $ ZH(Z \rightarrow e^+e^-, H \rightarrow $ inv) signal at the CEPC, with $ \sqrt{s} $ = 240 GeV, BR($ H \rightarrow $ inv) = 10% and integrated luminosity of 5.6 ab-1.

    4.   Result of upper limit and the boson mass resolution (BMR)
    • After the event selections, the 95% CL upper limit of BR($ H \rightarrow $ inv) is computed with CL$ _{s} $ formalism using the profile likelihood ratio as a test statistic [18] in which systematic uncertainties are ignored. The likelihood ratio method uses $ \mu $S+B, where $ \mu $ is the signal strength, S is the signal and B is the background. First, signal and background samples are fitted to obtain their distribution functions, which are used to generate Asimov data. And the Asimov data can provide a simple method to obtain the median experiment sensitivity of the measurement as well as fluctuations about this expectation. Then, construct the test statistic distribution generated for signal+background and background-only hypotheses assuming a signal strength $ \mu $, and each $ \mu $ is corresponding to a CL$ _{s} $ value calculated by the ratio of the two hypothesis probabilities [19]. When the CL$ _{s} $ is 0.05, the $ \mu $ value is its 95% CL upper limit. The corresponding negative logarithmic profile likelihood ratio -$ \Delta $log(L) as a function of $ \mu $ is shown in Fig. 5. The horizontal axis corresponding to -$ \Delta $log(L) = 2 on the y-axis is approximately 95% CL interval of $ \mu $.

      Figure 5.  The $\mu$ distribution from likelihood profile, where the dashed lines indicate the location of approximately 68%, 95% CL interval, which corresponds to -$\Delta$log(L) = 0.5, 2 on the y-axis.

      Table 6 summarizes the expected precision on the measurement of BR($ H \rightarrow $ inv) and the 95% CL upper limit on BR($ H \rightarrow $ inv) with a dataset of 5.6 ab-1. The combined 95% CL upper limit of three channels is estimated to be 0.26%. Any evidence of invisible Higgs decays that exceeds this value will indicate BSM phenomenon.

      $ZH$ final Precision on Upper limit on
      states BR($H \rightarrow$ inv)$\times$100 (%) BR($H \rightarrow$ inv) (%)
      $Z \rightarrow e^+e^-$, $H \rightarrow$ inv 45.37 1.08
      $Z \rightarrow \mu^+\mu^-$, $H \rightarrow$ inv 23.57 0.55
      $Z \rightarrow q\overline{q}$, $H \rightarrow$ inv 9.54 0.27
      Combination 8.68 0.26

      Table 6.  Expected precision on the measurement of BR($H \rightarrow$ inv) and the 95% CL upper limit on BR($H \rightarrow$ inv) with a dataset of 5.6 ab−1.

      The precision of the upper limit will be affected by various systematic uncertainties [20], such as the luminosity, the beam energy, the efficiency of the object reconstruction, and the acceptance of the detector. The precision of luminosity can achieve 0.1%, and the beam energy is expected to be better than 1 MeV which can be ignored on experimental recoil mass measurements. For tracks within the detector acceptance and transverse momenta larger than 1 GeV, a track finding efficiency is better than 99%. These systematic uncertainties are expected to be small and will be ignored in this paper.

      The precision of the upper limit of BR($ H \rightarrow $ inv) in $ ZH(Z \rightarrow qq, H \rightarrow $ inv) channel strongly relies on the invariant mass reconstruction of Z boson. And the boson mass resolution (BMR) is defined as the visible invariant mass resolution of the $ ZH(Z \rightarrow qq, H \rightarrow $ inv) event to quantify the invariant mass reconstruction of Z boson.

      The BMR of CEPC detector can reach 3.8% under ARBOR reconstruction algorithm [10, 21]. A fast simulation is performed to quantify this dependence. The fast simulation takes into account the signal of $ ZH(Z \rightarrow qq, H \rightarrow $ inv) and the main background of $ ZZ (Z \rightarrow qq, Z \rightarrow $ inv) after event selection. Fig. 6 shows the dependence of the accuracy [16] of $ ZH(Z \rightarrow qq, H \rightarrow $ inv) and different BMR. For the BMR between 4% and 20%, the accuracy degrades rapidly as BMR increases, and for the BMR less than 4%, the change of accuracy is less than 0.06%. Through the fast simulation, it can be concluded that BMR is vital and will affect the measurement precision of the $ ZH(Z \rightarrow qq, H \rightarrow $ inv) channel. Therefore, the BMR less than 4% will be an essential reference for detector design and optimization.

      Figure 6.  This figure indicates the dependence of the accuracy of the qqH (H$\rightarrow$inv) channel on the BMR under the background of $ZZ(Z \rightarrow qq, Z \rightarrow$ inv). The dashed lines show the accuracy when BMR is 2%, 3.8%, 6%, and 20% under assuming the BR($H \rightarrow$ inv) = 10%.

    5.   Conclusion
    • This paper studied the measurement of Higgs invisible decays at the CEPC, and the upper limit of Higgs invisible decays is measured with three independent channels $ ZH(Z \rightarrow qq, H \rightarrow $ inv), $ ZH(Z \rightarrow \mu^+\mu^-, H \rightarrow $ inv), and $ ZH(Z \rightarrow e^+e^-, H \rightarrow $ inv). The combined result of 95% CL upper limit of BR($ H \rightarrow $ inv) is 0.26% for three channels. Comparing with the LHC results, which are 26% from ATLAS and 19% from CMS, the result of the CEPC will be improved by two orders of magnitude. Comparing with High-Luminosity LHC (HL-LHC) result at 14 TeV, which is expected to 2.5% [22], the result of CEPC will be improved as one order of magnitude. The accuracy of the upper limit of the CEPC is significantly better than the hadron colliders, because the reconstructed Higgs recoil mass spectrum at the electron-positron Higgs factories gives a very clear and distinct signature of the Higgs boson, as well as the high productivity of the Higgs bosons at the CEPC. Comparing with other electron-positron colliders, like the International Linear Collider (ILC) and the Future Circular Collider (FCC-ee), which the 95% CL upper limit of BR($ H \rightarrow $ inv) is 0.26% for ILC [23] and 0.22% for FCC-ee (5 ab-1 at 240 GeV and 0.19% by combining 365 GeV) [24], the CEPC result is consistent with these results. Among all these three signal channels, the $ qqH $ channel gives the best result due to its largest number of events. The precision of upper limit in $ qqH $ channel strongly relies on the invariant mass of the visible hadronic decay system, and BMR better than 4% provides a clear separation between the Higgs signal and the $ ZZ $ background, which shall be pursued as one of the key physics requirement for the CEPC detector design in the future.

      The authors would like to thank Chengdong FU and Xianghu ZHAO for providing the simulation tools and samples. We also thank Patrick Janot for helpful discussion about FCC results.

Reference (24)

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