Cross Section and Higgs Mass Measurement with Higgsstrahlung at the CEPC

The Circular Electron Positron Collider (CEPC) is a future Higgs factory proposed by the Chinese high energy physics community. It will operate at a center-of-mass energy of 240-250 GeV. The CEPC will accumulate an integrated luminosity of 5 ab$^{\rm{-1}}$ in ten years' operation, producing one million Higgs bosons via the Higgsstrahlung and vector boson fusion processes. This sample allows a percent or even sub-percent level determination of the Higgs boson couplings. With GEANT4-based full simulation and dedicated fast simulation tool, we evaluated the statistical precisions of the Higgstrahlung cross section $\sigma_{ZH}$ and the Higgs mass $m_{H}$ measurement at the CEPC in the $Z\rightarrow\mu^+\mu^-$ channel. The statistical precision of $\sigma_{ZH}$ ($m_{H}$) measurement could reach 0.97\% (6.9 MeV) in the model-independent analysis which uses only the information of Z boson decay. For the standard model Higgs boson, the $m_{H}$ precision could be improved to 5.4 MeV by including the information of Higgs decays. Impact of the TPC size to these measurements is investigated. In addition, we studied the prospect of measuring the Higgs boson decaying into invisible final states at the CEPC. With the standard model $ZH$ production rate, the upper limit of ${\cal B}(H\rightarrow \rm{inv.})$ could reach 1.2\% at 95\% confidence level.

On the other hand, many new physics models predict the Higgs couplings deviate from the SM at the percent level. Thus the percent or even sub-percent level precision becomes necessary for the future Higgs measurement program. However, this accuracy is difficult to achieve at the LHC [9]. Moreover, as the Higgs boson can only be reconstructed through its decay products at the LHC and thus it is impossible for the LHC to access the Higgs total width or absolute couplings in a model-independent way.
Compared to the hadron collider, an electron positron collider has significant advantages in the precision measurements of the Higgs boson. The beam energy and polarization of the initial states are precisely known and adjustable. Thus the Higgs production cross section is available with the recoil technique. In this way, a lepton collider can provide the absolute measurements of Higgs couplings [10][11][12]. Besides, it is free of the QCD backgrounds. Almost every Higgs event can be recorded and reconstructed. Therefore, an electron-positron Higgs factory is an essential step in understanding the nature of the Higgs boson.
The Circular Electron Positron Collider is a Higgs factory proposed by the Chinese high energy physics community [12]. It will operate at a center-of-mass energy of 240-250 GeV with an instantaneous luminosity of 2 × 10 34 cm −2 s −1 . With two detectors operating over 10 years, the CEPC will accumulate about one million Higgs events, corresponding to an integrated luminosity of 5 ab −1 .
The SM Higgs bosons are produced via the processes of e + e − → ZH (Higgsstrahlung), e + e − → ννH (W W fusion) and e + e − → e + e − H (ZZ fusion) at the CEPC [13][14][15][16][17][18], as shown in Fig. 1. The corresponding production cross sections for the SM Higgs boson of 125 GeV, as functions of center-of-mass energy, are plotted in Fig. 2. At the center-of-mass energy of 250 GeV, the Higgs bosons are dominantly produced from ZH process, where the Higgs boson is produced in association with a Z boson. Feynman diagrams of the Higgs production mechanisms at the CEPC: the Higgsstrahlung, W W fusion, and ZZ fusion processes. The branching ratio of the Z boson decaying into a pair of muons is 3.3%. The muons can be easily identified and their momentum can be precisely measured in the detector. By tagging the muon pairs from the associated Z boson decays, the Higgsstrahlung events can be reconstructed with the recoil mass method: where E µ + and E µ − are the energies of the two muons, M µ + µ − is their invariant mass, and s is the square of center-of-mass energy. Therefore, the ZH (Z → µ + µ − ) events form a peak in the M recoil distribution at the Higgs boson mass.
With the recoil mass method, the ZH events are selected without using the decay information of the Higgs boson. Thus the inclusive ZH cross section σ ZH and the coupling g HZZ can be determined in a modelindependent manner. The measured g HZZ , combined with exclusive Higgs boson decay measurements, could be used to determine the Higgs boson width and absolute values of couplings between the Higgs boson and its decay final states [19]. Meanwhile, the Higgs mass m H can be extracted from the M recoil distribution. A good knowledge of the Higgs mass is crucial since the m H is the only free parameter in the SM Higgs potential and it determines the Higgs decay branching ratios in the SM. Based on the model-independent analysis, the Higgs decay information can be used to further suppress the backgrounds, leading to a better m H precision.
The recoil mass method allows better exclusive measurement of Higgs decay channels. Many new physics models predict a significant branching ratio of the Higgs boson decaying to invisible products [20][21][22][23]. At the LHC, the current upper limit of this branching ratio is about 40% [24,25], which is much larger than the value predicted in the SM (B(H → inv.) = B(H → ZZ → νννν) = 1.06×10 −3 ). At the CEPC, this measurement can be significantly improved by using the recoil mass method. In this paper, we evaluate the upper limit on the branching ratio of the Higgs decaying to invisible final states.
A series of simulation studies of similar processes have been performed at the International Linear Collider (ILC) [10,26]. Compared to the ILC, the collision environment of the CEPC is significantly different. The ILC uses polarized beams while the CEPC has no beam polarization. Besides, the beam spot size of the CEPC at the interaction point (IP) is much larger than that of the ILC, leading to a much weaker beamstrahlung effect and a narrower beam energy spread [10,12,27]. The details of parameter comparison are listed in Table 1 [27]. Due to the above differences, the cross sections for both signal and backgrounds are different. Therefore, it is necessary to perform the full detector simulation at the CEPC. This paper is organized as follows. Section 2 describes the detector model, Monte Carlo (MC) simulation and samples used in the studies. Section 3.1 presents the measurements of ZH cross section and the Higgs mass 010201-2 in a model-independent manner. The dependencies of measurement precisions on the TPC radii are investigated in Section 3.2, providing a reference for the future detector optimization. The model-dependent analysis of the Higgs boson mass is described in Section 3.3, and the measurement of the Higgs decaying to invisible final states is presented in Section 3.4. In Section 4, we discuss the systematic uncertainties and the methodology of systematic control. The conclusion is summarized in section 5.

Detector and Monte Carlo Simulation
The analysis is performed on the MC samples simulated on the CEPC conceptual detector, which is based on the International Large Detector (ILD) [28,29] at the ILC [10]. With respect to the ILD, the CEPC conceptual detector has a L * (the distance between the interaction point and QD0, the final focusing magnet) of 1.5 m, which is significantly shorter than that of the ILC (4.5 m). The shorter L * is essential to achieve a high luminosity by reducing the beam nonlinearity in the interaction region. Besides, the CEPC has multiple interaction points, thus the push-pull operation is not necessary. Therefore, the thickness of return Yoke is reduced by 1 meter at the CEPC conceptual detector.
Apart from the L * and return Yoke, the CEPC conceptual detector follows the same design of the ILD. Installed in a solenoidal magnet of 3.5 Tesla, the CEPC conceptual detector consists of a vertex detector, a tracking system and a calorimetry system. The silicon pixel vertex detector (VTX) consists of three cylindrical and concentric double-layers, with an innermost radius of 16 mm [28,29]. The tracking system is composed of a time projection chamber (TPC) as the main tracker and the silicon tracking devices, including a silicon inner tracker (SIT), forward tracking disks (FTDs), a silicon external tracker (SET) and end-cap tracking disks. The VTX and SIT are expected to provide a spatial resolution of better than 3 µm near the interaction point, which is crucial for the vertex reconstruction and the jet flavor tagging. The outermost FTD disk is positioned at z = 1057.5 mm to the IP. With an inner radius of 92.7 mm and an outer radius of 309 mm, it improves the geometric acceptance of the tracking system to |cosθ| <0.995. The TPC has nearly 200 three-dimensional (r, φ and z) spacepoints, with inner and outer radii of 0.325 m and 1.8 m respectively, and a half-length of 2.35 m. It provides an expected spatial resolution of better than 100 µm in the rφ plane. The SET provides a precise position measurement outside the TPC. Such a tracking system is expected to achieve a precise determination of the charged particle momenta with a resolution of where p T is the transverse momentum and θ is the polar angle. The calorimetry system is composed of a high granularity electromagnetic calorimeter and a high granularity hadron calorimeter, allowing excellent separations of showers of different particles. It is expected to provide a jet energy resolution of 3-4% and enable the PID efficiency to be over 99.5% for muons with momentum larger than 10 GeV. More information about the CEPC conceptual detector can be found in reference [12]. A set of event samples at a center-of-mass energy of 250 GeV, corresponding to an integrated luminosity of 5 ab −1 , has been generated with Whizard 1.95 [30,31]. It consists of the SM Higgs signal with m H = 125 GeV and the major SM backgrounds, including the γγ process (photon-induced background e + e − → e + e − γγ → e + e − l + l − , where the photons are generated according to the Weizsäcker-Williams approximation [32][33][34]), 2fermion processes (e + e − → ff , where ff refers to all lepton and quark pairs) and 4-fermion processes, categorized as ZZ, W W , ZZ or W W , single Z (e + e − Z) and single W (e + ν e W or e −ν e W ). If the final states could be produced through both W W and ZZ intermediate states, such as e + e − ν eνe , this process is classified as "ZZ or W W " and their interference is included. The initial state radiation (ISR) is also taken into account in the sample generation. More details about the CEPC samples can be found in reference [35].
The Higgs signal samples are fully simulated with Mokka [36] and reconstructed with ArborPFA [37]. A beam energy spread of 0.16% has been included in this analysis.
In order to save the computing power, a fast simulation framework has been developed to process the backgrounds. In the fast simulation, the detector responses are obtained by a series of full simulations for single particle events. Then the responses, including momentum resolution and detection efficiency, have been parameterized as functions of energy and polar angle for different types of particles. The fourmomenta of the visible final state particles are smeared according to the parameterized resolutions and they are randomly accepted based on the corresponding detection efficiencies.

The Analyses
The expected number of ZH events N ZH can be expressed as where L is the integrated luminosity, B(H → X) is the branching ratio of an exclusive Higgs decay mode, and (H → X) is the corresponding selection efficiency. In the model-independent analysis using only information in the Z boson decays, the efficiencies are expected to be uniform for each Higgs decay mode, we can write and thus σ ZH can be determined in a model-independent manner. The Higgs decay information can be used to further suppress the SM backgrounds and improve the precision on the mass measurement. In this case, the selection efficiency (H → X) depends on the Higgs decay mode.

Model-independent analysis on σ ZH and m H measurement
In the model-independent analysis, the event selection is composed of a pre-selection and a multivariate analysis (MVA). In the pre-selection, a pair of oppositely charged muons is required. The pair with the minimum [38]. The invariant mass of µ + µ − is required to satisfy 80 GeV < M µ + µ − < 100 GeV. In order to suppress 2-fermion backgrounds, the transverse momentum of the muon pair, p Tµ + µ − , is required to be larger than 20 GeV and the difference of the azimuth angles of the two muons should be less than 175 • . The Toolkit for Multivariate Analysis (TMVA) [39] is used to the further background rejection. In this paper, the method of gradient Boosted Decision Trees (BDT) is adopted and the selected variables for TMVA input are M µ + µ − , p Tµ + µ − , the polar angle of Z candidate, and the acollinearity of muon pair, which is defined as where p µ ± is the momentum vector of µ ± . After the pre-selection, a half of the remaining backgrounds are selected for training, together with another copy of signal sample of 5 ab −1 . The BDT response is calculated using weights obtained from training samples and applied to the whole data set, shown in Fig. 3. With the requirement of BDT>-0.05, the signal/background ratio is improved from 12.3% to 31.1%. The BDT selection is optimized to the σ ZH measurement. The cut flow is summarized in Table 2 and the signal selection efficiency is 62.8%. After the selection, the leading backgrounds are from ZZ (18.8% of the remaining background), γγ (21.8%) and 2-fermion (32.8%) processes. The selected muons may also come from the ZH events with the Z boson not decaying to µ + µ − . About 200 events of this type survive after the event selection and they are flatly distributed in the signal region. They are neglected due to their small contribution (∼ 0.29%) in the total background.  The final recoil mass spectrum of µ + µ − is shown in Fig. 4. An unbinned maximum likelihood fit to the M recoil distribution is performed in the region of 120 GeV to 140 GeV to determine the signal yield as well as the value of the Higgs mass. The background is represented by a third order Chebychev polynomial function, whose parameters are fixed to the values extracted from the background samples. The Higgs signal shape is described by a Crystal Ball function. Based on the fit results, σ ZH is estimated to a relative precision of 0.97% and m H to a precision of 6.9 MeV. The uniformity of event section efficiency with different Higgs decay modes is studied. A SM ZH (Z → µ + µ − ) sample corresponding to 500 ab −1 has been simulated, where the Higgs boson decays inclusively. Fig. 5 shows the efficiencies and no significant bias to any specific Higgs decay mode is observed. In order to evaluate the impact of the sensitivities to the various Higgs decay modes, the SM ZH cross section is kept unchanged and a specific SM Higgs decay branching ratio is enlarged by 5% each time (B(H → X) → B(H → X) + 5%)). All branching ratios are then scaled to keep the total event rate and the resultant differences in the σ ZH measurement are summarized in Table 3. The largest bias in the σ ZH is less than 10 −3 , which is much smaller than the statistical uncertainty 0.97%. It is therefore reasonable to conclude the recoil mass method gives a model-independent measurement.  From the detector point of view, the precisions of σ ZH and m H are mainly determined by the detector solid angle acceptance and the muon identification efficiency. Besides, the m H precision also relies on the muon momentum resolution. The momentum resolution scales approximately with the inverse of BL 2 , where B and L represent the magnetic filed strength and the detector radius respectively. The tracking system of the CEPC conceptual detector is composed of a silicon tracking system and a main tracker of TPC. A larger TPC radius, corresponding to a larger lever arm, will give a better momentum resolution but it brings more construction cost. The performances at different TPC radii are studied using the fast simulation tool. In the modelindependent analysis, the expected accuracies of σ ZH and m H are recorded, see Fig. 6. If the TPC radius is reduced by 25%, the precisions of σ ZH and m H are worsened by 2% and 20%, respectively. These expected accuracies are then parameterized as functions of the TPC radius. For the σ ZH measurement, it is expressed as where δσ ZH σ ZH (%) is the relative precision of cross section measurement and R TPC (m) is the TPC radius. Similarly, the accuracies of Higgs mass δm H at different TPC radii are obtained. Its dependence on the TPC radius can be expressed as

Model-dependent analysis on m H measurement
Assuming the SM Higgs decays, the background can be further suppressed by using the Higgs decay information, leading to a better Higgs mass measurement. On top of the pre-selection criteria used in the modelindependent analysis, we request that there are more than four charged tracks reconstructed. In the MVA stage, the energy of all reconstructed final states, E vis , is also taken as an input variable except those in the MVA of model-independent analysis. After the final selection, the recoil mass spectrum is shown in Fig. 7 and an efficiency of 66.1% is obtained. The resultant precision of m H is improved to 5.4 MeV.

The measurement of the invisible decay mode of the Higgs boson
The invisible decay mode of the Higgs boson is a well motivated signature of the physics beyond the SM [20][21][22][23]. At the CEPC, the invisible Higgs boson decay branching ratio can be determined precisely using the recoil mass method. Assuming the Higgs boson has the SM coupling to the Z boson and non-vanishing couplings to the beyond SM invisible particles, the measurement potential of the Higgs boson decaying to invisible final states at the CEPC is investigated at different values of B(H → inv.).
The signal candidates are identified using the same pre-selection as that in the model-independent analysis. In addition, we require that there is no extra visible energy except that of the muon pair decayed from Z boson and that E vis must be within 105 GeV and 125 GeV. Fig. 8 shows the µ + µ − recoil mass spectrum of the candidates with B(H → inv.) = 50%. In this scenario, the final signal event selection efficiency is 63.9% and the relative precision of the cross section of Higgs decaying to invisible final states δσ ZH,H→inv. /σ ZH,H→inv. reaches 1.16%.  Based on different assumptions of B(H → inv.), the relative precisions of δσ ZH,H→inv. /σ ZH,H→inv. are given in Fig. 9. The upper limit of B(H → inv.) at 95% confidence level is estimated to be 1.2 × 10 −2 by using the likelihood ratio test method [40].

Discussion on systematic uncertainties
A complete investigation of potential systematic uncertainties is beyond the scope of this paper. Here we present several sources of systematic uncertainties and the strategies to deal with them. With the best knowledge, The common uncertainties on σ ZH and m H measurements include differences between the data and the MC simulation for the tracking efficiency and PID, which can be investigated and corrected using a high purity control sample of about 20 M Zγ (ISR return) events.
In the measurement of cross sections, the important uncertainties are from ISR correction factor (1+δ, defined as the ratio of observed cross section over the Born one), luminosity measurement, the branching fraction of intermediate state decay (Z → µ + µ + ), as well as fitting procedure.
The uncertainty of 1 + δ depends on the precisions of both the experimental line shape measurement on the Born σ ZH below 250 GeV and the theoretical radiator function. We expect the latter will be calculated to a precision that is negligible comparing to the statistical uncertainty by the time of the CEPC data taking. The former is estimated with Born cross sections at six centerof-mass energies equally distributed from the threshold to 250 GeV. The luminosity is set at 50 f b −1 for each energy point below 250 GeV and the cross sections are generated according to the formula in Ref. [35] with statistical uncertainties. The generated cross sections are fitted with the same formula, but the coupling is free and the Higgs mass is float in one standard deviation according to PDG [38]. Then the resultant line shape is used to calculate the 1 + δ. We repeat the generation, fitting, and calculation procedure 1000 times, and the spread of 1+δ is determined to be 0.1% and taken as the systematic uncertainty of correction factor.
The integrated luminosity could be measured using small angle radiative Bhabha scattering and the expected precision is better than 10 −3 . The current uncertainty in the B(Z → µ + µ − ) is 0.2% [38], which will be further improved by the Z boson samples at the CEPC. The uncertainty of fitting procedure could be estimated by changing the background shape and fitting range, and the difference in the measured σ ZH is taken as the systematic uncertainty.
In the Higgs mass measurement, the dominant uncertainty may be from beam energy measurement. In order to control this uncertainty to MeV level, the new technology needs to be developed to improve the precision of beam energy measurement. Another potential uncertainty is the mass shift between the measured Higgs mass and the truth value. In order to control the shift, the dependence of mass shift on the Higgs mass input is investigated around 125 GeV. Then it is extracted as a third order Chebychev polynomial function. The measured Higgs mass is corrected by the fit function. The combined uncertainty of fit function and the remaining shift, 1.5 MeV, is taken as the systematic uncertainty.
The consistence between the fast and full simulation is checked using the ZZ events with at least one pair of muons found. The invariant mass of µ + µ − is shown in Fig. 10. In the concerned region between 80 GeV and 100 GeV , the statistics of the full-simulated ZZ sample is 2.32% lower than that of the fast-simulated. If the remaining background is reduced by 2% after the final event selection, the precision of σ ZH is varied from 0.974% to 0.971% and the m H precision varied from 6.91 MeV to 6.87 MeV. Therefore this effect can be safely ignored in this analysis. From the above discussions, the systematic uncertainty should be under control while the statistical one will be dominated at the CEPC.

Summary
The CEPC is expected to play a crucial role in understanding the nature of the Higgs boson. In this paper, the statistical precisions of Higgs production cross section σ ZH and mass m H measurements at the CEPC are investigated with full simulated Higgsstrahlung signal of 5 ab −1 integrated luminosity at the centerof-mass energy of 250 GeV. Using the recoil mass method, the statistical precision of σ ZH could reach 0.97%, corresponding to a 0.49% accuracy of g HZZ . The expected statistical accuracy of m H is 6.9 MeV while it is improved to 5.4 MeV with inclusion of the Higgs decay information. The dependence of these results on TPC radius is investigated and parameterized. Reducing the TPC radius by 25%, the statistical precisions of σ ZH and m H are worsened to 0.98% and 8.4 MeV, respectively. In addition, we explored the potential of the invisible decay mode of the Higgs boson at the CEPC. The upper limit of B(H → inv.) at the 95% confidence level could reach 1.2 × 10 −2 . All above results are incorporated into the CEPC-SPPC Preliminary Conceptual Design Report [12].
The same measurement is studied at the ILC in the Z → e + e − and Z → µ + µ − channels with an integrated luminosity of 2 ab −1 and polarized beams of P (e − , e + ) = (−0.8, 0.3) [10,26]. The g HZZ precision could reach 0.4% while the upper limit of B(H → inv.) is 1.7 × 10 −2 . For the m H measurement, the current m H precision is 0.24 GeV achieved at the LHC [8] and it will be improved to 50 MeV at the HL-LHC [9]. At the ILC, the statistical precision of m H could reach 14 MeV [10,26]. Compared with these facilities, the improvement at the CEPC is due to weaker beamstrahlung and higher statistics.