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

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2.Detector design and Monte Carlo simulations

One of the physics programs at the CEPC is precision measurement of the Higgs boson properties. The CEPC detector is expected to reconstruct and identify all key physical objects, including charged leptons, photons, jets, missing energy, and missing momentum.
The CEPC-v4 [6] detector was designed 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 composed of a tracking system, Time-Projection-Chamber tracker (TPC), high granularity calorimeter system, solenoid that generates a 3 Tesla magnetic field, and muon detector embedded in a magnetic field return yoke. The tracking system consists of silicon vertexing and tracking detectors. The calorimetry system consists of an electromagnetic calorimeter (ECAL) and an iron scintillator for a 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 the center-of-mass energy of 240 GeV, corresponding to the overall luminosity of 5.6 ab^?1, are generated with WHIZARD1.95 [15]. The generated events are then processed with MokkaC, and an attempt is made to reconstruct every visible particle with ARBOR [10]. The cross sections of the major SM processes of the $ e^{+}e^{-} $ collisions as functions of the center-of-mass energy $ \sqrt{s} $ are used in the simulations, including the Higgs boson 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 using Geant4 based full detector simulations and reconstruction. Limited by finite computational resources, approximately 20% of the two fermion backgrounds are used in full simulations.
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 $ e^{+}e^{-}\rightarrow ZH $, which is called the “$ ZH $” process. Then, Z bosons decay into leptons or hadrons, and the Higgs particle decays into two Z bosons, which eventually decay into four neutrinos. For the backgrounds, the major SM backgrounds are divided into 2-fermion processes and 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 pairs, 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. 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, are named “single_z”. The “single_w” processes include one electron, one electron neutrino, and two other fermions. If a final state includes a pair of electrons and a pair of electron neutrinos simultaneously, the corresponding processes are named “szorsw”. In addition to the above-mentioned backgrounds, the same four fermions in the final state can be combined into different two intermediate bosons. If the two intermediate bosons can be two Z bosons, the processes are named “zz”. The “ww” process is the one in which two intermediate bosons can become two W bosons. If two intermediate bosons can become two Z bosons or two W bosons, the corresponding process is “zzorww”.

Type	Four final states
single_z	Two electrons, two other fermions or two electron neutrinos, and two other fermions
single_w	One electron, one electron neutrino, and 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

Table1.Six types of 4-fermion backgrounds.

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

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

In the $ ZH(Z \rightarrow qq, H \rightarrow $ inv) process, owing to the presence of quarks, many final states are expected. The event selection uses the information about all visible particles, and the distributions of the signal and backgrounds are shown in Fig. 2. The comprehensive event selections are as follows. In the $ e^+e^- \rightarrow ZH(Z \rightarrow qq, H \rightarrow $ inv) process, the mass of the system recoiling against all visible particles from the Z boson is $ M_{\rm{recoil}}^{\rm{visible}} $, which can be calculated using 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, $ 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 between the azimuthal angles of the two jets should be below 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

Figure2. (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 the subsequent cuts (based on Table 3). The blue arrows mark the cut ranges.

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

(2)

where $ M_{\rm{recoil}}^{\rm{visible}} $ is approximately 125 GeV, $ \sqrt{s} = 240 $ GeV, and the invariant mass of the 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). Using the values of the parameters in Eq. (2), $ 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. Owing to the presence of quarks, the final states may include many charged particles. It is necessary to limit the number of charged particles ($ N_{\rm{charged}} $) with energy greater than 1 GeV to be 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} $) should be less than 95 GeV, for suppressing the backgrounds containing tau and quarks.
Table 3 lists the yields of signal ($ 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. The compositions of these backgrounds are similar to the signal channel and are difficult 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%

Table3.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.

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

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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 the $ 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 for background suppression. The event selections are as follows. The recoil masses of two muons ($ M_{\rm{recoil}}^{\mu^{+}\mu^{-}} $) and two electrons ($ M_{\rm{recoil}}^{e^{+}e^{-}} $) can be calculated using Eq. (1). The peak of $ M_{\rm{recoil}}^{\mu^{+}\mu^{-}} $ and $ M_{\rm{recoil}}^{e^{+}e^{-}} $ distribution should be around the Higgs boson mass 125 GeV. Considering the resolutions of muons and electrons and the distributions 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^{-}} $) is closer to the Z boson mass. To suppress the 2-fermion backgrounds, the transverse momenta of the muon pair ($ P_{\rm{T}}^{\mu^+\mu^-} $) and the electron pair ($ P_{\rm{T}}^{e^{+}e^{-}} $) are required to be more than 12 GeV, as shown in Fig. 3(c) and Fig. 4(c). Moreover, the angle between the two muons ($ \Delta\phi_{\mu^+\mu^-} $) should be under 175° or that between the two electrons ($ \Delta\phi_{e^+e^-} $) should be under 176° to suppress the 2-fermion backgrounds. The visible energy ($ 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 the Z boson decays; the value of $ E_{\rm{visible}} $ is approximately 105 GeV, as shown in Fig. 3(d) and Fig. 4(d). Using the approximate values of $ M_{\mu^{+}\mu^{-}} $ and $ E_{\mu^{+}\mu^{-}} $ in the relativistic energy-momentum relation $ M_{\mu^{+}\mu^{-}}^2 = E_{\mu^{+}\mu^{-}}^2- P_{\mu^{+}\mu^{-}}^2 $, the value of $ {E_{\mu^{+}\mu^{-}}}\big/{P_{\mu^{+}\mu^{-}}} $ is close to 2, similar to that of $ {E_{e^+e^-}}\big/{P_{e^+e^-}} $.

Figure3. (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 the subsequent cuts (based on Table 4). The blue arrows mark the cut ranges.

Figure4. (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 the subsequent cuts (based on Table 5). The blue arrows mark the cut ranges.

Table 4 lists the yields for the $ \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 difficult to suppress further. The remaining backgrounds containing the muon, tau, and two neutrinos account for 38% of the total backgrounds. The algorithm TAURUS does not increase the significance of the $ \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%

Table4.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 lists the yields for the $ eeH $_inv signal and its backgrounds at the CEPC. The cut Vertex$ _{\tau}< $0.0011, which is the position of the decay vertex, changes the value of significance from 10.91 to 13.79. Since the signal channel does not contain tau, the Vertex$ _{\tau} $ of the signal channel is much smaller than the 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 about tau particles cannot further suppress these backgrounds. In conclusion, the tau-finding algorithm TAURUS can improve the significance of the 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%

Table5.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.Results for the upper limit and the boson mass resolution (BMR)

$ZH$ final states	Precision of BR($H \rightarrow$ inv)$\times$100 (%)	Upper limit on BR($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

5.Conclusion

This paper studied the measurement of the Higgs invisible decays at the CEPC. The upper limit on the Higgs invisible decays was measured using 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 for the 95% CL upper limit of BR($ H \rightarrow $ inv) was 0.26% for the three channels. Compared with the LHC results (26% for ATLAS and 19% for CMS), the result obtained at the CEPC is better by two orders of magnitude. Compared with the High-Luminosity LHC (HL-LHC) result for 14 TeV, which is expected to be 2.5% [22], the result obtained at the CEPC is better by one order of magnitude. The accuracy of the upper limit for the CEPC is significantly better than those for 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.
The CEPC result is consistent with the results for other electron-positron colliders, such as the International Linear Collider (ILC) and the Future Circular Collider (FCC-ee), for which the 95% CL upper limits on BR($ H \rightarrow $ inv) were 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]. Among these three signal channels, the $ qqH $ channel yielded the best result owing to its largest number of events. The precision of the upper limit on the $ 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 requirements for designing future CEPC detectors.
The authors would like to thank Chengdong FU and Xianghu ZHAO for providing the simulation tools and samples. We also thank Patrick Janot for a helpful discussion about FCC results.

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

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

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