The $Higgs\to b\bar{b},c\bar{c},gg$ measurement at CEPC

This analysis is based on a centre-of-mass energy of $250\,GeV$ and an integrated luminosity of $5\,ab^{-1}$, which is slightly different from the normal setting with a centre-of-mass energy of $240\,GeV$ and an integrated luminosity of $5.6\,ab^{-1}$. The signal events have two isolated leptons, $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$, mostly from the Z-boson decay in the Higgs-strahlung process. The invariant mass and recoil mass of these two leptons should be close to the Z boson and the Higgs boson, respectively. The signal events also have two jets generated from the Higgs boson decay. Jet kinematics, i.e. the invariant mass and angle of the two jets, is used to improve the separation performance between signal and background.

After event selection, a template fitting method is used to determine the component fractions of the $H\to b\bar{b}$, $H\to c\bar{c}$, and $H\to gg$ processes. The relative accuracy of the $H\to b\bar{b}/c\bar{c}/gg$ signal strength, corresponding to the centre-of-mass energy of $250\,GeV$ and an integrated luminosity of $5\,ab^{-1}$, is 1.1%/10.5%/5.4% in the $\mu^{+}\mu^{-}H$ channel and 1.6%/14.7%/10.5% in the $e^{+}e^{-}H$ channel. The relative accuracy in the $\mu^{+}\mu^{-}H$ channel is better than that in the $e^{+}e^{-}H$ channel because, first, the background in the $e^{+}e^{-}H$ channel is significantly larger than that in the $\mu^{+}\mu^{-}H$ channel due to the single-Z processes, and second, the momentum resolution for $\mu^{\pm}$ is better than that for $e^{\pm}$. Extrapolating to the CEPC nominal settings under the assumption that the signal efficiencies and background suppression rates are the same at the two centre-of-mass energies, the $H\to b\bar{b}/c\bar{c}/gg$ signal strength accuracy is 1.57%/14.43%/10.31% in the $e^{+}e^{-}H$ channel and 1.06%/10.16%/5.23% in the $\mu^{+}\mu^{-}H$ channel.

This subsection describes the measurement of the branching fractions of $H\to b\bar{b}/c\bar{c}/gg$ in the $\nu\bar{\nu}H$ channel. The first step focuses on the separation of Higgs-to-two-jets signal events from the entire sample corresponding to SM prediction with a cut-based event selection and the TMVA tool TMVA . The cut variables are designed according to the characteristics of the signal and background, which are described below, and the cut flow is summarized in table 1. The $\gamma\gamma$ is the abbreviation for the $\gamma\gamma\to hadrons$ process.

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  Most of the $\nu\bar{\nu}H$ events are from ZH process with $Z\to\nu\bar{\nu}$, while the WW-fusion contributes about 13% (ignore the interference when calculating the contribution of WW fusion to $\nu\bar{\nu}H$). The signal events have only the jets from the Higgs boson decay. Therefore, the signal should have a recoil mass peak at the mass of the Z boson. However, the SM process of $\nu\bar{\nu}Z(Z\to q\bar{q})$ is an irreducible background for this analysis. In the recoil mass distribution (see the left plot of figure 5), the signal and the $\nu\bar{\nu}Z(Z\to q\bar{q})$ backgrounds peak at the mass of the Z boson, and the other SM backgrounds peak at two sides of the distribution. An optimized cut on the recoil mass has a signal efficiency of 88% and reduces the background by more than one order of magnitude, see the first row in table 1. The signal has significant invisible energy, so the visible energy (visEn, the second row in table 1) is about half of the total energy. The leptonic and semi-leptonic backgrounds have high-energy leptons in the final state, and the fully hadronic backgrounds have a larger multiplicity in the final state. So, the cut variable of the leading lepton energy (leadLepEn, the third row in table 1) is used to suppress leptonic and semi-leptonic backgrounds, the cut variable of multiplicity (the fourth row in table 1) is used to suppress leptonic and some fully hadronic backgrounds. With the above cuts, 0.01% of the leptonic backgrounds, 9.24% of the semi-leptonic backgrounds and 4.82% of the fully hadronic backgrounds remain in the selected sample.

  

  

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  The remaining backgrounds are dominated by 2f processes consisting of $e^{+}e^{-}\to q\bar{q}$. The $e^{+}e^{-}\to q\bar{q}$ backgrounds with high-energy Initial State Radiation (ISR) detected by the detector would have energetic neutral particles in the final state. Meanwhile, the final state particles of $e^{+}e^{-}\to q\bar{q}$ would fly in the end-cap region, so the visible transverse momentum ($Pt$, the sixth row in table 1) would be lower and the visible longitudinal momentum ($Pl$, the seventh row in table 1) would be higher than in the signal events. With the cut variables of the leading neutral energy (leadNeuEn, the fifth row in table 1), $Pt$, and $Pl$, 0.3% of the $e^{+}e^{-}\to q\bar{q}$ backgrounds remain in the selected sample.

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  A cut on Y23⁴⁴4The Durham distance at which a two-jet system can be reconstructed into a three-jet system. Catani:1991hj ; Cacciari:2011ma is applied to suppress the backgrounds, resulting in an improvement of the signal strength accuracy from 0.94% to 0.76%, see the eighth row in table 1. The signal should have an invariant mass (InvMass, the ninth row in table 1) close to that of the Higgs boson. After applying the above cut variables, the distributions of invariant mass for signal and backgrounds are shown in the right plot in figure 5. The $\nu\bar{\nu}Z(Z\to q\bar{q})$ backgrounds have their peak at the Z boson, the other SM backgrounds exhibit a flat distribution with statistics comparable to those of $\nu\bar{\nu}Z(Z\to q\bar{q})$ and signal. The cut variable of the invariant mass could preserve 75% of the remaining signal events and veto 69% of the remaining backgrounds.

After the cut-based event selection, a Boosted Decision Tree (BDT) is implemented to further improve the selection performance. The input variables include the cut variables mentioned above and the four-momentum of two jets. Figure 6 shows the BDT responses. With the optimized cut of the BDT response at 0.02, over 71% of the backgrounds are rejected at the cost of a 10% signal loss. With a signal efficiency of 34%, the first step reduces background by more than four orders of magnitude, resulting in a relative accuracy of 0.47% in the measurement of $\nu\bar{\nu}H,H\to q\bar{q}/gg$. The selection efficiency for $H\to b\bar{b}/c\bar{c}/gg$ is 35%/33%/35%. The inhomogeneity in $H\to b\bar{b}/c\bar{c}/gg$ selection is mainly caused by the cut variables Y23 and invariant mass, which are discussed in section 5.

The second step focuses on the separation of the different Higgs decay modes and can be divided into two stages. The first stage aims to obtain the optimized flavor tagging performance matrix (see below) of the CEPC baseline detector, and the second stage calculate the signal strength accuracy.

In the first stage, the particles in the final state are forced into two jets by using the Durham Catani:1991hj jet clustering algorithm implemented in the LCFIPlus software package LCFIPlus . The jet clustering algorithm considers single reconstructed particles and composite objects such as reconstructed secondary vertices as basic candidates. For each jet, the flavor tagging algorithm is used to calculate its likeness to reference samples of b or c jets. The flavor tagging used in this work is also implemented in the LCFIPlus software package and is performed using Gradient Boosted Decision Tree (GBDT). The training is applied to the simulated $Z\to q\bar{q}$ sample produced at $\sqrt{s}$ of $91.2\,GeV$. The reconstructed jets in the sample are divided into 4 categories depending on the number of secondary vertices and isolated leptons in the jet: jets with secondary vertex and lepton, jets with secondary vertex but without lepton, jets without secondary vertex but with lepton and jets without secondary vertex and lepton. In each category, two types of flavor tagging algorithms are trained using the GBDT method, one for the b-tagging algorithm and the other for the c-tagging algorithm.

The distributions of b/c-likeness are shown in figure 7 for $\nu\bar{\nu}H(H\to b\bar{b})$, $\nu\bar{\nu}H(H\to c\bar{c})$, $\nu\bar{\nu}H(H\to gg)$, and the remaining SM backgrounds. The phase space spanned by the b/c-likeness is divided into three different regions corresponding to the identified b, c, and gluons. We then obtain the ratio of b-jet identified as b-jet, b-jet identified as c-jet, and so on. These ratios can be represented with a migration matrix, the form of which is shown in figure 7. We optimize the working point (phase space separation) to maximize the trace of the migration matrix. The optimized migration matrix is shown as the bottom plot in figure 7. In principle, the working point can be optimized independently for $H\to b\bar{b}$, $c\bar{c}$, and $gg$ measurements. We evaluate the corresponding performance and find that the final accuracy can be improved by sub-percent level. Since the improvement is not significant, a uniform matrix for $\nu\bar{\nu}H(H\to b\bar{b}/c\bar{c}/gg)$ is used for simplicity. According to the identified jet-flavor combinations, the signal events and backgrounds are classified into six different categories (see figure 8).

In the second stage, the relative accuracy of the signal strength could be calculated by the log-likelihood function (1) Zyla:2020zbs ; MLE ,

where $S_{b}$ represents the signal strength of $\nu\bar{\nu}H(H\to b\bar{b})$, $N_{b,i}$ represents the event count of $\nu\bar{\nu}H(H\to b\bar{b})$ in the $ith$ bin, $N_{bkg,i}$ represents the event count of the backgrounds in the $ith$ bin, and $N_{i}$ represents the total event count ($\nu\bar{\nu}H$ with $H\to b\bar{b}/c\bar{c}/gg$ and backgrounds) in the $ith$ bin, similar for $S_{c}$, $S_{g}$, $N_{c,i}$, and $N_{g,i}$. The error covariance matrix is obtained from the Hessian matrix of the log-likelihood function with respect to three signal strengths. The relative accuracies of the signal strengths are the square roots of the diagonal elements of the covariance matrix. It is 0.49%/5.75%/1.82% for $\nu\bar{\nu}H(H\to b\bar{b}/c\bar{c}/gg)$.

In this subsection, the accuracy of the $q\bar{q}H(H\to b\bar{b}/c\bar{c}/gg)$ signal strength is analyzed. The analysis process is similar to that in the $\nu\bar{\nu}H$ channel. Since the backgrounds consist of leptonic, semi-leptonic, and fully hadronic samples, the first step can be divided into three stages to select the signal events step by step.

1. 1.

   Selection of fully hadronic events.

2. 2.

   Selecting events with 4-jet topology.

3. 3.

   Selection of ZH events.

The cutflow corresponding to these three stages is given in table 2. The $\gamma\gamma$ is the abbreviation of $\gamma\gamma\to hadrons$ process. The first stage aims to suppress the leptonic and semi-leptonic backgrounds that have low multiplicity or high-energy leptons ($e^{\pm}/\mu^{\pm}$) or invisible leptons ($\nu/\bar{\nu}$). Thus, with the cut variables of multiplicity (the first row in table 2), the leading lepton energy (leadLepEn, the second row in table 2), and the visible energy (visEn, the third row in table 2), the background statistic is reduced to 26%. The cut variable of visible energy can also veto some $e^{+}e^{-}\to q\bar{q}$ backgrounds with high-energy ISR that have escaped the detector. Step to the second stage: the cut variables of the leading neutral energy (leadNeuEn, the fourth row in table 2, aims to suppress $e^{+}e^{-}\to q\bar{q}$ with high-energy ISR detected by the detector), thrust⁵⁵5To evaluate the thrust of an event, first determine the thrust axis $n_{T}$, which is the direction of maximum momentum flow. The thrust is then defined as the fraction of the particle momentum that flows along the thrust axis.  ATLAS:2020vup (the fifth row in table 2), and Y34 ⁶⁶6The Durham distance at which a three-jet system can be reconstructed into a four-jet system. (the sixth row in table 2), are used to select 4-quark samples from the full hadronic samples. The second stage reduces the remaining background by almost an order of magnitude at the cost of losing 11% of the signal events.

Since the signal contains only four jets, the particles in the final state are first forced into four jets using Durham algorithm. There are two bosons in the signal, then the four jets in the final state are paired into two di-jet systems using the pairing method of the minimization eq. (2),

where $M_{12}$ and $M_{34}$ are the masses of the di-jet systems and $M_{B1}$ and $M_{B2}$ are the reference masses of the Z or W or the Higgs boson. The $\sigma$ is the convolution of the boson width and the detector resolution. According to CEPCStudyGroup:2018ghi , the detector resolution is 4% of the boson mass. After pairing four jets into two di-jet systems, we refer to the di-jet system with heavy invariant mass ($M_{heavy}$) as the heavy di-jet system and the one with light invariant mass ($M_{light}$) as the light di-jet system. There are angular variables that can be used to separate the signal from the remaining backgrounds. Angular variables include: the angle between two jets of a light di-jet system (ZJetsA, the seventh row in table 2), the angle between two jets of a heavy di-jet system (HiggsJetsA, the eighth row in table 2), and the angle between two di-jet systems (ZHiggsA, the ninth row in table 2). These three cut variables could reduce more than 84% (from $3.97\times 10^{7}$ to $6.10\times 10^{6}$) of the backgrounds. For the signal, one di-jet system should have an invariant mass near the Higgs boson and the other should have an invariant mass near the Z boson. Then, a circular selection (the tenth row in table 2) $(M_{heavy}-125)^{2}+(M_{light}-91)^{2}<=29^{2}$ can be used to select signal events, as shown in figure 9.

To fully exploit the characteristics of the signal and backgrounds, a BDT method is used to suppress the backgrounds. Input variables include the cut variables mentioned above, the four-momentum of four jets, and several event shape variables ATLAS:2020vup including max-broadening⁷⁷7The max-broadening is related to the transverse momentum measured with respect to the thrust axis. Introduce a plane perpendicular to the thrust axis and divide the space into two hemispheres. The jet broadening of each hemisphere is defined as $B=\frac{1}{2\sum_{j=1}^{N_{particles}}|P_{j}|}\sum_{i=1,P_{i}\cdot n_{T}>0}^{N_{particles}}|P_{i}\times n_{T}|$, where $P_{i}$ is the 3-momentum of particle i, $n_{T}$ is the thrust axis, and $P_{i}\cdot n_{T}>0(P_{i}\cdot n_{T}<0)$ is used to divide the particles into two hemispheres. The max-broadening is then the maximum jet broadening between these two hemispheres., C-parameter⁸⁸8The linearized sphericity is defined as $L^{ab}=\frac{1}{\sum_{j=1}^{N_{particles}}|P_{j}|}\sum_{i=1}^{N_{particles}}\frac{P_{i}^{a}P_{i}^{b}}{|P_{i}|}$, where $P_{i}$ is the 3-momentum of particle i, and $P_{i}^{a}$ denotes the component a of the 3-momentum of the particle i. Then the C-parameter can be calculated as $C=3(\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{3})$, where $\lambda$ is the eigenvalue of $L^{ab}$., and D-parameter⁹⁹9The D-parameter can be calculated as $D=27\cdot\lambda_{1}\cdot\lambda_{2}\cdot\lambda_{3}$. . Finally, the total SM background is reduced to 1.07 million statistics, and more than 36% of the total $q\bar{q}H(H\to q\bar{q}/gg)$ signal events survived, resulting in a relative uncertainty of 0.57%.

After the event selection process, an optimized flavor tagging performance matrix can be found by setting an optimized working point on the distributions of b/c-likeness of two jets from the heavy di-jet system, which is shown in the top plot of figure 10. Compared to $\nu\bar{\nu}H$, the diagonal elements have decreased by 13%/8%/10% for $b/c/g$. In other words, the identification performance of the $b/c/g$ jet in the $q\bar{q}H$ channel is slightly worse than that in the $\nu\bar{\nu}H$ channel. This is due to the visible particles decaying from the Z boson would degrade the jet clustering performance. Based on the optimized flavor tagging performance matrix, the identified flavor combinations of $q\bar{q}H(H\to b\bar{b})$, $q\bar{q}H(H\to c\bar{c})$, $q\bar{q}H(H\to gg)$, and backgrounds are shown in figure 10. The x and y axes represent the flavor of two jets from the heavy and light di-jet systems, respectively. The light di-jet system in the signal events corresponds to the Z-boson. Due to the imperfect flavor tagging performance, the pattern of Z-boson decay in figure 10 is not clear. Using the log-likelihood function similar to that of $\nu\bar{\nu}H$, the relative accuracy for $q\bar{q}H(H\to b\bar{b}/c\bar{c}/gg)$ is calculated to be 0.35%/7.74%/3.96%.

To first order, the relative accuracy of $H\to b\bar{b}/c\bar{c}/gg$ signal strength is measured independently in three channels, $\nu\bar{\nu}H$, $q\bar{q}H$, and $\ell^{+}\ell^{-}H$. At a centre-of-mass energy of $240\,GeV$ and an integrated luminosity of 5.6 $ab^{-1}$, the relative statistical accuracy of $H\to b\bar{b}/c\bar{c}/gg$ signal strength can reach 0.27%/4.03%/1.56% when combined with these three channels. The results of the analysis are summarized in table 3. According to the recently released Snowmass Gao:2022lew , the CEPC operates at $240\,$GeV will integrate $20\,ab^{-1}$ luminosity. Accordingly, the relative statistical accuracy of the $H\to b\bar{b}/c\bar{c}/gg$ signal strength would be 0.14%/2.13%/0.82%.

The flavor tagging performance can be described by the migration matrix, which is defined by the bottom plot in figure 7. We have three reference points for the migration matrix: the unitary matrix corresponding to perfect flavor tagging performance, the flat matrix (all elements are equal to one-third) corresponding to the performance without flavor tagging, and the matrix corresponding to the CEPC baseline detector, which is shown as the bottom plot in figure 7.

An interpolation method is used to obtain different flavor tagging performance matrices, shown as the eq. (3),

where $M_{I}$ represents the perfect flavor tagging performance matrix (identity matrix), $M_{1/3}$ represents the matrix without flavor tagging (all elements equal $1/3$), $M_{opt}$ represents the flavor tagging performance of the CEPC baseline detector, $Tr_{I}$ represents the trace of the perfect flavor tagging performance matrix, and $Tr_{1/3}$ represents the trace of the matrix without flavor tagging. $Tr_{mig}$ is a variable whose value and the temporary matrix ($M_{mig}$) have a one-to-one relationship. The value of $Tr_{opt}$ ranges from $Tr_{1/3}$ to $Tr_{I}$. If $Tr_{mig}$ is greater than $Tr_{opt}$, use the upper formula of eq. (3). Else we use the lower formula. The value of $Tr_{mig}$ is varied from 1.0 (without flavor tagging) to 3.0 (perfect flavor tagging) in increments of 0.1. The dependence of signal strength accuracy on the flavor tagging performance is shown in figure 11. Accuracies corresponding to the baseline CEPC detector are represented by the markers at $Tr_{mig}$ = 2.34. With an ideal flavor tagging algorithm, the signal strength accuracy is 0.48%/3.53%/1.61% for $\nu\bar{\nu}H(H\to b\bar{b}/c\bar{c}/gg)$, which is a 2%/63%/13% improvement over the baseline CEPC detector (0.49%/5.75%/1.82%). The performance of the flavor tagging depends on the design of the vertex detector. If the key vertex detector parameters, including the inner radius, material budget, and spatial resolution, are changed by a factor 0.5/2 VTX from the baseline design (the geometry we used in this simulation), the $Tr_{mig}$ value changes accordingly from 2.35 to 2.54/2.16, as shown in figure 11.

Similar to $\nu\bar{\nu}H$, the dependence of the $q\bar{q}H(H\to b\bar{b}/c\bar{c}/gg)$ signal strength accuracy on the flavor tagging performance is shown in figure 12. With perfect flavor tagging performance, the relative accuracy is 0.26%/3.48%/1.41% for $q\bar{q}H(H\to b\bar{b}/c\bar{c}/gg)$, which is a 35%/122%/181% improvement over the baseline CEPC detector (0.35%/7.74%/3.96%). There is a significant improvement for $q\bar{q}H(H\to gg)$, because after event selection the backgrounds consist mainly of the processes of $e^{+}e^{-}\to q\bar{q}/W^{+}W^{-}/ZZ/Mixed$ with c-jets/light-jets in the final state, while the CEPC baseline performance classifies almost all light-jets and 30% of c-jets as gluon-jets. When the key vertex detector parameters, including inner radius, material budget, and spatial resolution, are changed by a factor 0.5/2 VTX from the baseline design, the $Tr_{mig}$ value changes accordingly from 2.12 to 2.31/1.93, as shown in figure 12.

The performance of the CSI can be evaluated by the angles between the reconstructed bosons and the MC truth bosons, $\alpha_{1}$ and $\alpha_{2}$, as shown in figure 13. Since the CSI evaluator used in this paper uses the MC truth information, it is only a demonstrator to illustrate the importance of an excellent CSI reconstruction.

After the entire event selection in table 2, the distributions of $log_{10}(\alpha_{1})$ versus $log_{10}(\alpha_{2})$ are shown in figure 14. The circle $(log_{10}(\alpha_{1})+3)^{2}+(log_{10}(\alpha_{2})+3)^{2}=11.11$ can improve the signal-to-background ratio. The distribution of $(log_{10}(\alpha_{1})+3)^{2}+(log_{10}(\alpha_{2})+3)^{2}$ for the signal and backgrounds is shown in figure 15, which shows that most backgrounds have relatively poor CSI performance compared to the signal events. This is because the backgrounds with good CSI performance were strongly suppressed by the event selection. For example, the right plot in figure 15 shows the distributions of $(log_{10}(\alpha_{1})+3)^{2}+(log_{10}(\alpha_{2})+3)^{2}$ for the samples of $e^{+}e^{-}\to W^{+}W^{-}\to 4\ quarks$, where the red line corresponds to all samples, while the blue line corresponds to that after event selection. To illustrate the performance of the CSI, each of these two distributions is normalized to a unit area. We can see that only the backgrounds with poor CSI performance passed the event selection process. An ideal CSI performance evaluator such as the quantity shown in the left plot of figure 15 shows a potential to improve the relative accuracy of $q\bar{q}H(H\to b\bar{b}/c\bar{c}/gg)$ signal strength by 6%/77%/90%. This motivates future developments aimed at improving the CSI reconstruction or developing a performance estimator based on reconstructed quantities.

Subsections 4.1 and 4.2 quantify the impact of flavor tagging performance on objective measurement, which strongly promotes better flavor tagging performance. Better flavor tagging performance can be pursued by optimizing the vertex detector and developing advanced reconstruction algorithms. The CEPC vertex detector is designed as a barrel-shaped structure with three concentric cylinders of double-sided layers, whose parameters are listed in table 4. The main features of the vertex detector are a single-point resolution of the first layer of better than $3\,\mu m$, a material budget of less than 0.15% X₀ per layer, and the location of the first layer near the beam pipe with a radius of $16\,$mm. The flavor tagging algorithm used in this analysis is implemented in the LCFIPlus package and is based on a GBDT.

For the optimization of the vertex detector, a previous exercise VTX quantifies the correlation between flavor tagging performance and relevant detector properties, including inner radius, material budget, and spatial resolution of the vertex system, as shown in figure 16. This exercise shows that significant improvements can be achieved if the above properties can be reduced compared to the baseline design. Considering an optimal and a conservative scenario, as proposed by ref. VTX , where the number of these three parameters is 0.5/2 times compared to the baseline design (the geometry we used in this simulation), the $Tr_{mig}$ will be changed from 2.35 to 2.54/2.16 in the $\nu\bar{\nu}H$ channel, as shown in figure 11, and from 2.12 to 2.31/1.93 in the $q\bar{q}H$ channel, as shown in figure 12. The details can be found in appendix B.