Detector geometry

The DC is surrounded by silicon detectors in both the barrel and end-cap regions, covering the radial range of 800–1,800 mm and the Z range of ±2980 mm. The detailed parameters of the DC are listed in Table 1.

A small cell design is selected to obtain a sufficient number of track hits at the outer radius. The ratio of the half-height to the half-width (the distance between a sense wire and its neighboring field wire in the r-ϕ plane) is approximately 1; thus, the shape of the drift cell is almost square. The drift cell contains a sense wire surrounded by eight field wires. In total, there are 25,357 drift cells of size 18 mm×18 mm. To solve the ambiguity [32] caused by multiple hits in the same layer, the drift cells between neighboring layers are offset by the half-width.

Because stereo wires have technical advantages in measuring the positions of charged tracks in the Z-direction, the DC is composed purely of stereo wires. All stereo wires are organized into 55 coaxial layers. The stereo angle on each layer varies as 0.028–0.062 rad, and the wires on any two neighboring layers are tilted in opposite directions.

Materials with low densities and atomic numbers are primarily considered to minimize multiple scattering. Both the inner and outer cylinders are made of carbon fibers. The working gas for the DC is a mixture of helium and C₄H₁₀ at a ratio of 90:10. The sense wire is made of gold-plated tungsten with a diameter of 20 μm, and the field wire is made of silver-plated aluminum with a diameter of 40 μm.

The DC is described using DD4hep, and the detector geometry parameters, such as dimensions, materials, and sensitive detectors, are stored in XML(eXtensible Markup Language)[33] format. XML serves as a compact file for DD4hep and the syntactic structure of the compact XML description is obtained from the SiD [34] detector description. Therefore, the detector design is associated with a set of corresponding XML files. By versioning the changes in the set of XML files, the geometric version can be easily controlled.

In the simulation, the DC is constructed by parsing the geometric parameters from the XML files using a specialized detector constructor, creating individual components, and placing them accurately in the correct positions to form a complete virtual detector. The construction process is as follows. A cylindrical-shaped volume is created according to the maximum and minimum radii and length of the DC. Subsequently, within the holy volume, wire layers with thicknesses of 18 mm are created at different radii, each of which is represented by a hollow cylinder. For a specific layer, the sense and field wires are directly constructed as tubs and placed at the right locations. The field wires are shared between neighboring cells with each field wire assigned to one drift cell as an entire tube. Because of the large number of wires, if every drift cell is created independently, it deteriorates the performance in terms of both speed and memory. To solve this problem, an alternative method is implemented to first construct the layers and then divide each layer into drift cells by utilizing DDSegmentation[30] from DD4hep.

After the geometrical model for the DC is built, the Geant4 visualization tool can be used to produce graphical representations of the detector and draw views and sections of the detector. Figure 4a shows a visualization of the sense and field wires created in the simulation, illustrating the r-ϕ projection of the proportion of the first ten layers of wires. The sense wires of each layer forms a rotating hyperboloid surface and Fig. 4b shows the sense wires in the first layer. With the help of the visualization tool, the parameters of all individual geometrical entities can be displayed and checked. Careful validation of the geometry lays a solid foundation for the next steps in digitization and tracking.

Fig. 4Full-screen View

(Color online) (a) Projection of the first 10 layers of the DC in the r-ϕ plane. The red dots represent sense wires, while the green dots represent field wires. (b) Visualization of sense wires in the first layer. The skew is exaggerated

Digitization

The DC is used to measure the spatial coordinates of the trajectory of a charged particle. This is achieved by detecting the ionization electrons produced by the charged particles in the gas of the DC and measuring their drift time and arrival positions on the sense wires.

In the context of detector simulation, digitization refers to the simulation of the detector response. For the DC, an accurate simulation of the drift time is particularly important. Because the CEPC detector R&D is in its early stages, a detailed detector design is still not available. A simplified digitization method is implemented to support the development of the tracking algorithm. Its workflow is illustrated in Fig. 5a.

Fig. 5Full-screen View

(a) The specific workflow of digitization. (b) Schematic diagram of the doca

In the simulation using Geant4, a small step size i.e 0.5 mm) is selected to simulate the passage of charged particles through the DC. When the particle enters the drift cell, the distance between each Geant4 step and the sense wire of the cell is recorded. The smallest distance, referred to as doca (the distance of closest approach), represents the closest approach of the particle trajectory to the sense wire as shown in Figure 5b, and is considered as the drift distance of this hit. The doca is smeared using a Gaussian function with a width equivalent to the wire resolution set to 110 μm. The drift time is calculated using the space-time relation (X-T relation) [35]. In this study, a relatively simple linear X-T relationship is used.

During digitization, the effects of various types of background events are considered. The hits from a signal event are overlaid with those from additional background events before the detector response was calculated. The background primarily originates from the minimum bias and beam–gas interactions. In this study, background events are assumed to be randomly distributed within a time window of 2,000 ns. For simplicity, track recognition is only performed on the group of the fastest hits, each of which has the minimum drift time among all hits belonging to a drift cell. Therefore, during the digitization stage, when a background hit precedes that of a signal within the same drift cell, the signal hit is overridden by the background hit. Furthermore, the default wire efficiency is assumed to be 100%, and all subsequent results are based on this assumption.

Track parameterization

In a homogeneous magnetic field, the motion of charged particles in the DC forms a trajectory that is described by the five parameters of a helix, as shown in Eq. 1. The first three parameters of the equation describe the circle on the X-Y plane, as shown in Fig. 6, and the last two parameters describe the information along the Z direction of the trajectory, following the conventions used by the BaBar experiment [37]. $\text{P}\equiv {\left(d_0\mathrm{,}{\varphi }_0\mathrm{,}\omega \mathrm{,}{z}_0\mathrm{,}\tan \lambda \right)}^{\text{T}}\mathrm{.}$ (1) The definition of each parameter is as follows:

• d₀ is the signed distance from the center of the projected circle of the helix to the reference point on the X-Y plane. Here, the interaction point (IP) is considered as the reference point.

• ϕ₀ is the azimuth angle of the center of the projected circle of the helix relative to the IP on the X-Y plane.

• ω is the curvature of the trajectory in the X-Y plane. Further, the sign represents the charge of the trajectory assigned by track fitting.

• z₀ is the signed distance between the helix and the IP along the Z direction.

• tan λ is the slope of the helix, that is, the inclination angle of the projection of the helix on the R-Z plane.

Because of material effects and magnetic field inhomogeneity, the real trajectory is not a perfect helix, causing these five parameters to change along the flight path of the trajectory. For the tracking algorithms, all DC hits are considered as measurement data. In this study, charged particles with transverse momentum greater than 0.8 GeV are used. This eliminates the possibility of multiple circular tracks, thus allowing the track to be accurately described by track parameters at the first DC hit. The track parameters are interconverted with the position and momentum of the hits by calling specific functions using the calculation formula referenced in [38].

Fig. 6Full-screen View

Description of the helix parameters (d₀, ϕ₀) and the position of the POCA (the point of closest approach)

Extension of geometry interfaces

The track reconstruction process considers the interaction with materials, such as energy loss and multiple scattering, which are highly dependent on the geometry, materials, and nonuniform magnetic field of the DC. To obtain these information, the following extended interfaces are developed by inheriting from GenFit. This because the tracking algorithm is based on the GenFit software package and a detailed explanation of GenFit is presented in Sect. 4.4:

• CEPCMagneticFieldProvider: This class inherits from AbsBField of GenFit, and provides the magnetic field strength at specific positions by integrating with DD4hep. It accepts the position as input and calculates the magnetic field strength from the magnetic field map of the detector.

• CEPCMaterialProvider: This class inherits from AbsMaterialInterface of GenFit, and obtains the geometry and material information of the DC based on GeomSvc.

Track finding

The mission of track-finding is to rapidly and accurately determine the potential DC hits produced by the same charged track based on candidate seed tracks [39]. DC track finding is implemented using three components, as shown in Figure 7: finding DC hits with CKF, salvaging DC hits, and appending SET hits.

Fig. 7Full-screen View

Workflow of track reconstruction

4.3.1

Hits finding

DC hit finding is among the main procedures in DC track finding, and it is implemented by reusing the CKF algorithm adopted from Belle II [40]. The CKF is an iterative local algorithm, first presented in [41], and widely used in HEP experiments. The CKF starts with a seed estimation of track parameters with uncertainties and then extrapolates the track into the detector volume using the Runge-Kutta-Nyström method [42].

In CEPCSW, the implementation of the CKF algorithm involves three main tasks: the acquisition of geometry information for the DC, input/output (I/O) management, and optimization of various parameters. The DC geometry can be obtained by implementing classes as introduced in Sect. 4.2. In addition, the I/O of the CKF algorithm exists in the Belle II data format, thus completing the conversion between EDM4hep and Belle II data model. Because of the different geometrical designs, working gas, and magnetic fields, the judgement criteria and standards of the CKF are optimized, and new thresholds are configured to adapt them to the CEPC DC, for example, the extrapolation track length, chi-squared(χ²) value, and doca.

In this study, five hypotheses are considered: electron, muon, pion, kaon, and proton. A schematic and brief overview of the track-finding process are illustrated in Fig. 8. The primary tracking process begins with seed tracks, based on which the search roads are built in the DC. These seeds are extrapolated along the search roads using the CKF, which attempts to smoothen the track while iteratively searching for neighboring hits both outward and inward. After extrapolation, potential candidate hits are identified based on the drift distance, current position, and uncertainties of the candidate track. When multiple candidate hits are found during the propagation step, a prediction is made for each candidate hit using the predicted track parameters and measured values to assess the next candidate hit. Subsequently, a new candidate hit is added to the track and the process is repeated. In cases involving multiple mutually exclusive next-candidate hits, the entire candidate track is duplicated and subsequently treated as two separate tracks. Finally, the final candidate track is selected based on various quality criteria, such as the residual distance between the extrapolated and measured hit positions, extrapolated track length, and the doca, etc.

Fig. 8Full-screen View

(Color online) Schematic of track finding process. For better visibility, the schematic shows only a few hits in the DC. First, determining the DC hits (the red dots) using CKF is based on the seed track (the blue curve), and a rough track (the red curve) is fitted using GenFit. Subsequently, the remaining hits are determined using the hits salvaging algorithm. By extrapolating the track to the remaining hits (the green dots), it is determined whether they belong to the track based on criteria, such as, the doca, residual of the extrapolated distance, etc.

This process produces a set of potential candidate hits, which are then further refined. The advantage of the CKF algorithm is its fast running speed. However, it is difficult to reliably determine all DC hits because the CKF algorithm uses restrictive conditions, leading to only one hit being found when multiple hits are present within the same layer. These conditions result in low track quality, particularly for layers with stereo wires and low-momentum particles. Therefore, a hit-salvaging algorithm is developed to solve this problem.

4.3.2

Hits salvaging and appending

The hit-salvaging algorithm serves as a supplement to the hit-finding algorithm and is developed independently to salvage more signal hits within the DC. This algorithm utilizes the preliminary results from the CKF, uses GenFit to perform rough track fitting, and extrapolates the track to the DC hits lost by the CKF.

During the examination, the hits are evaluated based on selection criteria, including the extrapolated track length and doca, to determine their association with the current track and to add them to the track. Subsequently, by refitting the previous track with the retrieved hits using GenFit, more precise track parameters can be obtained. These two steps can remove the majority of noise hits to achieve high hit purity.

Furthermore, the hit salvaging algorithm facilitates the enhancement of the overall accuracy and completeness of the track reconstruction process, improving track quality and hit efficiency (ϵ_hit). The hit efficiency is defined as Eq. 2: ${ϵ}_{\text{hit}}=\frac{N_{\text{find}}}{N_{\text{MC}}}\mathrm{,}$ (2) where N_find represents the number of found DC signal hits and N_MC represents the number of true DC hits. Figure 9 shows the hit efficiency as a function of p_T measured using single e^- events, which increased by approximately 26% at 2 GeV.

Fig. 9Full-screen View

Hit efficiency as a function of pT measured using single e^-, and the pink dots represent the hit efficiency with the hit salvaging algorithm, while the teal dots represent the hit efficiency without the hit salvaging algorithm

Next, the refined track is extrapolated to the SET plane using GenFit to determine the expected position of the SET hit. If the extrapolated hit position falls within 3σ of the actual hit position on the SET plane, the SET hit obtained by extrapolation is added to the track. This step enhances the accuracy and reliability of the candidate hits.

Track fitting

The track-fitting algorithm is performed to accurately determine the track parameters by fitting all the hits obtained in the aforementioned processes and to provide physics quantities such as momentum, vertex position, and error matrices of the track. Further, this enables the DC to exhibit good track resolution and kinematic discrimination capability [43, 44]. In the track fitting process, the momentum, vertex position, and error matrix of the track are obtained using the method of least squares[45]. The least squares method minimizes the χ² value to determine the parameter estimates for ${\chi }_{\text{min}}^2$. χ² is defined as follows. ${\chi }^2={{\displaystyle \sum _{i=1}^n\left(\frac{\text{drif}{t}_i-{\text{doca}}_i}{{\sigma }_i}\right)}}^2\mathrm{.}$ (3) where drift_i represents the fitted distance of the i^th hit on the track, σi represents the measured error of the drift distance for the i^th hit, and doca_i represents the distance between the i^th hit sense wire and the fitted track in space, that is, the doca. The fitting process involves multiple iterations to adjust the track parameters until the fit converges.

During the track-fitting procedure, the correct handling of measurement uncertainties and all physical effects disturbing the helical path of the track must be considered to arrive at a correct description of the track; these effects are highly dependent on the geometry of the detector and the non-uniform magnetic field. Considering all the physical effects renders the mathematical description of the track model very complicated and allows only a numerical calculation. Dedicated fitting algorithms based on GenFit are developed for this purpose because GenFit can accurately determine the track parameters of charged particles using spatial coordinate measurements provided by detectors. The GenFit package is integrated into CEPCSW by extending the necessary interfaces.

4.4.1

Extension of GenFit interfaces for CEPC

The configuration of GenFit necessitates a large amount of DC information, including the geometry parameters, detector materials, strength distribution of the magnetic field, position of the sense wires, and drift distance of hits, etc. Further, to facilitate the integration of GenFit into CEPCSW, seamless data conversion is necessary. Thus, the following GenFit interface is developed:

• CEPCToGenFitHitConverter: Extracting the necessary information based on the TrackerHit object [28] of EDM4hep, and creating a corresponding GenfitHit object which is used as the input of track fitting algorithm. It implements the data model conversion from EDM4hep to GenFit at the level of the hit.

• CEPCToGenFitTrackConverter: Storing the results of the track fitting algorithm, including general information (number of iterations, convergence, etc.), fit status (χ², NDF, p-value, track length, etc.), and constructing a Track object [28] as the output, to achieve data model conversion at the level of the track.

• WireMeasurementDC: Inheriting from WireMeasurement [23], which is used for measurements in wire detectors (such as straw tubes and drift chambers). To use this class, a plane must be described by the u and v axis with v coincident with the wire (and u orthogonal to it, obviously). This is because it is not valid for arbitrary choices of plane orientation.

• GenfitFitter: Inheriting from AbsKalmanFitter, which is used to call the fitter in the GenFit package, and enabling the configuration of the magnetic field, material effects, and fitter type.

The interfaces for obtaining material and magnetic field information are introduced in Sect. 4.2, and Fig. 10 shows the data flow during the track-fitting process.

Fig. 10Full-screen View

Data flow of track fitting process

After fitting, GenfitTrack objects contain a large amount of data: track representations, TrackPoint objects [23] with measurements, FitterInfo objects [23] that contain all fitter-specific information, and the fitting status used to assess the quality of the fitting. In addition, the fitted drift distance for each hit is obtained, which is useful for subsequent hit selection process. By analyzing this information, the tracks are reconstructed with high track efficiency.

Data samples

Two types of samples are produced to quantify tracking performance. The first sample encompasses five types of single particles (e^-, μ^-, π^-, K^-, and p) generated using the particle gun of Geant4. The second sample comprises physics events, specifically the e⁺e^- → ZH, $H\to {\mu }^{+}{\mu }^{-}$ process, which is generated using the Whizard [46] generator.

Each single-particle event is generated based on the specified polar angle θ and transverse momentum p_T, whereas the azimuthal angle ϕ is uniformly distributed from [-π,π). The barrel components of SIT and SET provide precise hit positions inside and outside the DC, improving the reconstruction efficiency, particularly for low-momentum charged particles. Therefore, single-particle events with cosθ < 0.776 correspond to the barrel region of the DC with a transverse momentum p_T in the range of [2, 50] GeV.

The $H\to {\mu }^{+}{\mu }^{-}$ process is a clean physics channel with a simple final state, which renders it suitable for testing the algorithm performance. The $H\to {\mu }^{+}{\mu }^{-}$ events are generated by Whizard at leading-order precision, considering the initial state radiation and final state radiation processes during event generation, as well as the 0.16% energy spread of the electron beam caused by beam-synchrotron radiation in CEPC.

Quality of track fitting

To ensure the reliability of track fitting, it is essential to perform a goodness-of-fit test, which typically includes two components: the χ² test and reduced residuals (pull) test. The χ² value is provided by GenfitTrack object after the fitting. The reduced residuals (pull) are defined as follows: $\text{pull}=\frac{{\nu }_{\text{fit}}-{\nu }_{\text{meas}}}{{\left[{\sigma }_{\text{fit}}^2-{\sigma }_{\text{meas}}^2\right]}^{\frac{1}{2}}}\mathrm{,}$ (4) where v_fit represents the fitted value obtained from the track reconstruction and v_mea is the measured value obtained from the simulation in the previous section. In addition, v_fit −v_meas is the residual between the fitted and measured values, and v_fit and v_meas denote the standard deviations of the fitted and measured values, respectively. The pull distribution conforms to a standard normal distribution with a good fit, indicating that most of the residuals are approximately zero and exhibit a symmetrical shape. The pull test is used to examine track momentum, drift distance, and track parameters. To calculate the pull, the true value is considered with a standard deviation of 0. Consequently, Eq. (4) is simplified to $\text{pull}=\frac{v_{\text{fit}}-v_{\text{meas}}}{{\left[{\sigma }_{\text{fit}}^2-{\sigma }_{\text{meas}}^2\right]}^{\frac{1}{2}}},$ (5) where P_fit expresses the fitted value and P_meas denotes the measured value.

The momentum resolution is an important metric in track reconstruction. Figure 11 shows the residual distribution between the transverse momentum of the fitted and truth tracks for single μ^- with 10 GeV. The distribution closely follows a Gaussian distribution, yielding a momentum resolution of 14.0 MeV, which satisfies the performance requirements of the CEPC tracking system.

Fig. 11Full-screen View

Blue dots represent the residual distribution between p_Trec and pTMC for single μ^- with 10 GeV. p_Trec and p_TMC represent the transverse momentum of the fitted track and the true track, respectively. The orange line represents the fitted Gaussian function

Spatial resolution is a critical standard that represents the accuracy of spatial position reconstruction. Figure 12 shows the residual distribution between the fitted and drift distances, which also exhibits a Gaussian distribution. The spatial resolution achieved is 106 μm, which is slightly below the set value of 110 μm. The method of fitting tracks containing measurement points [47], where the construction of χ² in the least-squares fit includes the contribution of the measurement points, can result in an underestimation of the spatial resolution compared to the true value in the reconstruction process.

Fig. 12Full-screen View

Blue dots represent the residual distribution between the fitted and drift distances for single μ^- with 10 GeV. The orange line represents the fitted Gaussian function

Track parameters resolution

The main results of the track reconstruction are the five track parameters as well as their distribution and resolution, which are related to the performance of the physics analysis. Owing to the influence of multiple scattering, correlations are observed between the track parameters, with significant correlations among the parameters within the same plane and smaller correlations between the parameters of two different planes. Furthermore, the geometric curvature κ (= 1/ω) of the track is related to transverse momentum p_T, where κ = q/p_T. Because the reconstructed instances use the same charge, the resolution of the transverse momentum p_T of the reconstructed track indirectly reflects the resolution of κ. The following resolutions for d₀, z₀, and transverse momentum p_T are provided to verify the results of the track reconstruction.

The resolution of the track parameters is obtained by fitting the distribution of the track parameters residuals using a Gaussian function. In the top panels of Fig. 13, the resolutions of the impact parameters d₀,z₀ and the relative resolution of p_T are shown as functions of the particle p_T. The bottom panels of Fig. 13 display the resolutions of the impact parameters d₀,z₀ at a fixed transverse momentum p_T = 8 GeV and the relative resolution of p_T as a function of the polar angle $\left|\cos \theta \right|$. For each p_T and $\left|\cos \theta \right|$, a sample of 20k events of single μ^- is generated. Owing to the increased material effect, the resolution of d₀, z₀ deteriorates at lower p_T and larger $\left|\cos \theta \right|$. The relative resolution of p_T is dependent on p_T because the curvature of the track on the X-Y plane is dominated by the multiple-scattering effect. In the high-momentum region, the momentum resolution is dominated by the single-point resolution of the tracks, resulting in worse resolution with an increase in p_T.

Fig. 13Full-screen View

Resolution of d₀ (left panels), z₀ (middle panels) and relative resolution of p_T (right panels) for single μ^- as function of particle p_T (top panels) and polar angle $\left|\cos \theta \right|$ (bottom panels) respectively

Tracking efficiency

Another important criterion for evaluating the track reconstruction performance is the tracking efficiency(ϵ), which is related to the tracking algorithm, particle type, and particle momentum. Before analyzing track efficiency, it is essential to establish a definition of a good track. A good track is defined as follows: the χ² value is below 400 and the number of DC signal hits on the track is more than six.

The definition of tracking efficiency ϵ is $ϵ\left(\%\right)=\frac{N_{\text{rec}}}{N_{\text{seed}}}\mathrm{,}$ (6) where N_seed is the number of satisfactory seed tracks successfully reconstructed using VXD and SIT, and N_rec is the number of reconstructed good-tracks.

As material effects easily lead to the generation of secondary particles, resulting in inconsistencies between the reconstructed track using MC particles as the seed and true tracks, high-quality track events are selected. When a charged particle enters the DC with minimal energy loss, it is considered to be a high-quality track, which is evaluated by $\left(p_{\text{mc}}-p_{\text{first}}\right)/p_{\text{mc}}$, where p_mc represents the true momentum and p_first represents the momentum of the first DC hit.

Figure 14 shows the track efficiency as a function of p_T for the five types of single particles, and the track efficiency for all types particles remains consistently above 99.5% without background, particularly for single μ^-, and the track efficiency is approximately 100%. Further, when a particle goes through the DC, only the energy loss of the particle is considered. If other interactions occur in the DC material and new types of particles are generated, these events are not considered in the tracking efficiency calculations. Thus, no distinction is made between particles and antiparticles in this study. The evaluation of track reconstruction with the background is important, as the track reconstruction of a real detector is susceptible to the background from different sources, significantly impacting the accuracy of the reconstruction result. To assess the performance of the tracking algorithm with the background and measure its ability to remove noise, the tracking efficiency is tested at different background levels. Figure 15 presents the track efficiency as a function of p_T for single μ^-, both without background and with a noise level of 20%. As evident, track efficiency is not significantly affected by the background, remaining consistently above 99.8%. This indicated the good stability and robustness of the tracking algorithm. The track multiplicity is a critical and significant issue, and preliminary research has been conducted. The generated MC events exhibit an average of 10 tracks with an average momentum of 2 GeV with a tracking efficiency of 99.6%. This suggests that the current track reconstruction algorithm can handle multiple tracks effectively and achieve good performance.

Fig. 14Full-screen View

Tracking efficiency as a function of pT measured using single particles, including e^-, μ^-, π^-, K^-, p, without background

Fig. 15Full-screen View

Track efficiency for single μ^- as a function of particle p_T. The light blue dots represent the results without background, and the light green dots represent the results with a noise level of 20%. For each p_T, a sample of 5k events is used for the study

Physics event reconstruction

To verify the accuracy of the tracking algorithm in physics events, e⁺e^- → ZH, $H\to {\mu }^{+}{\mu }^{-}$ process is analyzed. Figure 16 shows the reconstructed mass distribution of the Higgs boson fitted to the CrystalBall function. The mean value is measured at 125 GeV and aligned with the expected invariant mass of the Higgs boson. Further, the standard deviation is measured at 212.5 MeV.

Fig. 16Full-screen View

The reconstructed dimuon mass distributions from the $H\to {\mu }^{+}{\mu }^{-}$ events

Based on the aforementioned results and performance evaluations, it can be concluded that the tracking algorithm operates effectively and satisfies the performance requirements of the CEPC project.