ParticleNet and its application on CEPC Jet Flavor Tagging

A jet refers to a spray of stable particles formed through the hadronization of an energetic quark or gluon. The W/Z/Higgs boson and the top quark, the four most massive Standard Model (SM) particles, decay mainly into quarks and hadronize to jets. Figure 1 illustrates a reconstructed $e^{+}e^{-}\to Z\to c\bar{c}$ event with center-of-mass energy of 91.2 GeV. Efficient identification of the jet flavor could shed light on the properties of those massive particles and is critical for experimental exploration at the high-energy frontier. Flavor tagging is used to distinguish jets hadronized from different quark flavors or gluon. To promote the development of future electron-positron Higgs factories, which is regarded as the highest priority for the next collider ES , accurate performance analysis and optimization of both detectors and algorithms are essential. Jet flavor tagging and relevant benchmark analyses serve as good objectives.

The Circular Electron Positron Collider (CEPC) CEPCStudyGroup:2018ghi is a large-scale collider facility proposed after the discovery of the Higgs boson in 2012. It is designed with a circumference of 100 km with two interaction points. It can operate at multiple center-of-mass energies, including 240 GeV as a Higgs factory, 160 GeV for the $\rm W^{+}W^{-}$ threshold scan, and 91 GeV as a Z factory. It also can be upgraded to 360 GeV for the $t\bar{t}$ threshold scan. Table 1 summarizes its baseline operating scheme and the corresponding boson yields Sun:2023lut . In the future, it can be upgraded to a proton-proton collider to directly explore new physics at a center-of-mass energy of about 100 TeV. The main scientific objective of the CEPC is the precise measurement of the Higgs properties, especially its coupling properties. Additionally, trillions of $Z\to q\bar{q}$ events can provide an excellent opportunity for studying flavor physics.

Jet flavor tagging performance depends on detector design, particularly the design of the vertex detector, as well as the utilization of reconstruction algorithms. In this study, we apply ParticleNet Qu:2019gqs to the CEPC and assess its flavor tagging performance in the measurement of $\sigma(ZH)\cdot Br(Z\to\nu\bar{\nu},H\to c\bar{c})$ and $|V_{cb}|$ via W decay. Our results demonstrate that ParticleNet outperforms the baseline jet flavor tagging algorithm, LCFIPlus LCFIPlus , by achieving a 36% and 75% improvement in the relative statistical accuracy of $\sigma(ZH)\cdot Br(Z\to\nu\bar{\nu},H\to c\bar{c})$ and $|V_{cb}|$ measurement via W boson decay at the center-of-mass energy of 240 GeV and integrated luminosity of 5.6 $ab^{-1}$. We also observe that both ParticleNet and LCFIPlus perform better in the barrel region when compared to the endcap region. By analyzing the dependence of flavor tagging performance on vertex detector configurations, we observe that the most sensitive vertex detector parameter is the inner radius, followed by the material budget and spatial resolution. This result is consistent with previous studies conducted using LCFIPlus.

This article is organized as follows. Section 2 introduces the CEPC detector, software, and the samples used in this analysis. Section 3 describes the jet flavor tagging algorithms (LCFIPlus and ParticleNet) and the method used to evaluate their performance. Section 4 quantifies the dependence of flavor tagging performance on vertex detector configuration and compares the performance of ParticleNet and LCFIPlus. Finally, Section 5 provides a brief conclusion.

At present, two interaction points of CEPC are designed with the same baseline detector, which is designed according to the Particle Flow Algorithm (PFA) principle and emphasizes reconstructing visible final state particles in the most-suited detector subsystems. The structure of the CECP detector is shown in Fig. 2. From inner to outer, the baseline detector is composed of a silicon pixel vertex detector, a silicon inner tracker, a Time Projection Chamber (TPC) surrounded by a silicon external tracker, a silicon-tungsten sampling Electromagnetic Calorimeter (ECAL), a steel-glass Resistive Plate Chambers sampling Hadronic Calorimeter (HCAL), a 3 Tesla superconducting solenoid, and a flux return yoke embedded with a muon detector. For flavor tagging, the vertex detector is critical. At the CEPC, the vertex detector is designed with six concentric cylindrical layers of square silicon pixel sensors. The mechanical structure of the vertex detector consists of ladders, with each ladder supporting sensors on both sides. The detailed structure of the vertex detector is depicted in Fig. 3, and its specific parameters are listed in Table 2.

A baseline reconstruction software chain has been developed to quantify the scientific merit and guide the detector optimization of CEPC, see Fig. 4. The data flow of the CEPC baseline software starts from the event generators of Whizard whizard and Pythia pythia . The detector geometry is implemented into MokkaPlus mokka , a GEANT4-based full simulation module. MokkaPlus calculates the energy deposition in the detector-sensitive volumes and creates simulated hits. For each sub-detector, the digitization module converts the simulated hits into digitized hits by convolution of the corresponding sub-detector responses. The reconstruction modules include the tracking, the Particle Flow, and the high-level reconstruction algorithms. The digitized tracker hits are reconstructed into tracks via the tracking algorithms track . The Particle Flow algorithm, Arbor arbor , reads the reconstructed tracks and the calorimeter hits to build reconstructed particles. High-level reconstruction algorithms reconstruct composite physics objects such as converted photons, jets, taus, and so on, and identify the flavor of the jets.

In this paper, we utilized hadronic events at Z-pole operation, including 1 million $Z\to b\bar{b}$ events, 1 million $Z\to c\bar{c}$ events, and 0.33 million each of $Z\to u\bar{u}/d\bar{d}/s\bar{s}$ events. For ParticleNet, we divided the samples into three distinct sets: the training set for training the model, the validation set used to validate whether the model is overfitting or underfitting, and the testing set used to give flavor tagging results. The ratios of samples in these sets were set at 60%, 20%, and 20%, respectively. For LCFIPlus, we use all samples to do the test since we have already trained the model.

In this section, we introduce LCFIPlus and ParticleNet and compare their performance based on the CEPC detector and software. Both algorithms read the information of reconstructed jet candidates and calculate the jet likeness to b, c, and light categories.

The LCFIPlus package, a framework for jet analysis in linear collider studies, was originally developed by the International Linear Collider (ILC) ILC , and has since been widely used at the Compact Linear $e^{+}e^{-}$ Collider (CLIC) Robson:2018enq , the Future Circular Collider $e^{+}e^{-}$ (FCC-ee) FCCCDR , and CEPC. The LCFIPlus package consists of vertex finding, jet clustering, vertex refinement, and flavor tagging. To perform flavor tagging, the jets are classified into four categories based on the number of reconstructed vertices and isolated leptons in the jet. A set of variables is then extracted for each category, which includes the number of tracks in each vertex, the vertex mass, the distance between the secondary vertex and the primary vertex, the vertex decay length, the track transverse momentum, and more. Further details can be found in LCFIPlus . In each category, two types of flavor tagging algorithms are trained using the Gradient Boosted Decision Tree (GBDT) method, one for the b-tagging algorithm and the other for the c-tagging algorithm.

The ParticleNet based on Graph Neural Network (GNN) zhou2020graph was published at the beginning of 2019. The architecture of ParticleNet is shown in the left plot of Fig. 5. It consists of three EdgeConv blocks, one channel-wise global average pooling block, and two fully connected blocks followed by a softmax function to output the b/c/light-likeness for each jet. The core concept of ParticleNet is the EdgeConv operation, which is realized by applying feature aggregation for each particle and its k nearest particles in the jet. The specific process of each EdgeConv block is illustrated in the right plot of Fig. 5. It starts by finding the $k$-nearest neighbors for each particle within the jet. The edge between each particle and its $k$-nearest neighbors is determined using the input features of each particle. In the first EdgeConv block, the spatial coordinates ($\Delta\eta,\Delta\phi$) of the particles in the pseudorapidity-azimuth space are used to compute the edge of each pair of particles, while the subsequent EdgeConv blocks use the learned feature vectors as coordinates. The input features for our task, listed in Table 3, include the kinematic variables constructed with the 4-momentum of each particle, the PID information, the charge, and impact parameters. The distance between the interaction point and the path of a track is defined as the impact parameter, where the distance along the beam is called $z_{0}$ and perpendicular to the beam is called $d_{0}$.

Both flavor tagging algorithms assign three values to each jet: b-likeness, c-likeness, and light-likeness, with the constraint that their sum equals unity. The scatter plots in Fig. 6 show the distribution of b-likeness versus c-likeness for samples of $Z\to b\bar{b}/c\bar{c}$/light quarks with ParticleNet. In these plots, b-jets tend to concentrate in the region of larger b-likeness, c-jets in the region of larger c-likeness, and light-jets in the region of smaller b/c-likeness. The phase space spanned by the b/c-likeness is divided into three different regions corresponding to identified b, c, and light quarks. We then obtain the ratios of b-jets identified as b-jets, b-jets identified as c-jets, and so on. These ratios can be represented with a migration matrix, as shown in Fig. 7. The working point (phase space separation) can be optimized according to the specific analysis requirements. For general cases, we adopt the method using two orthogonal lines passing through the point (0.5, 0.5), as depicted by the two red lines in figure 6.

The performance of ParticleNet and LCFIPlus is evaluated by the following three criteria. The first one is the migration matrix since the perfect flavor tagging performance corresponds to the identity matrix. The second one is the physics performance, the better flavor tagging algorithm would induce better physics results. The last one is the vertex detector optimization since it is relevant to the resolution of transverse momentum and impact parameters, and further quantifies the reconstructed performance of the detector.

Flavor tagging, a methodology employed to discern the origins of jets, holds immense significance in the realm of experimental exploration at the High Energy Frontier. Jets originating from different quarks or gluons have key differences, represented in the multiplicity of different species of particles, the secondary vertices, the opening angle of jets, etc. The flavor tagging performance depends on both the flavor tagging algorithm and detector design. To pursue excellent discovery power and innovative design of the detector, intensive research and design towards the key detector technologies, especially towards the vertex detectors are performed. Meanwhile, the development of innovative algorithms injects new momentum into this field.

In this paper, we analyze the performance of ParticleNet and LCFIPlus. The ParticleNet based on GNN has been intensively used at CMS PRLQu ; PN1 ; PN2 and FCC-ee FCCPN . The LCFIPlus is a GBDT-based algorithm that has served as the baseline flavor tagging algorithm for CEPC and multiple future electron-positron Higgs factories. Using fully simulated hadronic events at a center-of-mass energy of 91.2 GeV at the CEPC baseline detector, we quantify the performance of both algorithms. We use a 3-dimensional migration matrix to describe the flavor tagging performance (representing the identification efficiency and misidentification rate), and the trace of the migration matrix is used as the key parameter to characterize flavor tagging.

At the CEPC baseline detector geometry, we observe that ParticleNet is significantly superior to LCFIPlus. At the inclusive hadronic Z pole sample, the trace of ParticleNet is larger than LCFIPlus by more than 14%. Consequently, the relative statistical accuracy of $\sigma(ZH)\cdot Br(Z\to\nu\bar{\nu},H\to c\bar{c})$ and $|V_{cb}|$ measurement via W boson decay is improved by 36% and 75%, respectively, when CEPC operates as a Higgs factory at the center-of-mass energy of 240 GeV and integrated luminosity of 5.6 $ab^{-1}$. Another paper Liao:2022ufk shows that ParticleNet can improve the statistical uncertainty of $R_{C}$ measurement by 60% at the CEPC. The flavor tagging performance, which is described by $Tr_{mig}$, of both ParticleNet and LCFIPlus depends on the polar angle. Both algorithms exhibit better performance in the barrel and smoothly degrade in the forward region. We also apply ParticleNet to different vertex detector geometries and observe that the flavor tagging performance is most sensitive to the inner radius, followed by the material budget and the spatial resolution. The result is consistent with that conducted by LCFIPlus. Benchmark performance in two scenarios, conservative and optimal, where the values of three vertex detector parameters are 2 and 0.5 times compared to the baseline design, reveals that ParticleNet can significantly enhance physics performance in the conservative scenario while showing less significant improvement with the aggressive detector design.