The STCF detector system is a sophisticated assembly designed to maximize physics potential in the τ-charm energy region. From the interaction point outward, it features a tracking system comprising an inner tracker (ITK) using radiation-hard technologies like μ RWELL-based gaseous detectors or MAPS-based silicon pixels, followed by a large main drift chamber (MDC) with helium-based gas for precise momentum measurement. PID is achieved via a barrel RICH detector and an endcap time-of-flight (DTOF) system, providing kaon-pion separation up to . The electromagnetic calorimeter (EMC) uses pure CsI crystals for high-resolution photon detection and energy measurement. A superconducting solenoid generates a magnetic field for tracking, surrounded by an iron yoke for structural support and flux return. The outermost layer is a muon detector (MUD) combining resistive plate chambers and plastic scintillators for efficient muon identification. The system is optimized for high luminosity, with advanced data acquisition handling event rates up to .

In this study, we utilize the Monte Carlo (MC) method with the KKMC and Tauola generators [17, 18] to generate events at a CME of based on the theoretical reaction cross-sections and branching ratios and simulate the electronic signals of the detector (including timing, amplitude, etc.) under a fast simulation package [19]. The process accounts for approximately (25%)²=6.2% (i.e., the signal fraction). The main background decays include [20]. The reconstruction software is then used to derive physical quantities such as the momentum and energy of the final state particles from these electronic signals.

The MC simulated data includes truth information (such as particle species) that is not available in actual experiments. This additional information helps optimize the parameter selection in the analysis algorithms, which in turn enhances the effectiveness of these algorithms when applied to real experimental data.

In the final state of the studied process, the detectable particles are , , and γ. The analysis algorithm filters events by reconstructing tracks in MDC, selecting those with exactly two charged tracks (corresponding to and ) and a total charge sum of zero. It is important to note that only tracks within a small distance from the electron-positron collision point are considered valid to eliminate cosmic rays and beam-related backgrounds. Further, the analysis algorithm uses PID based on ionization energy loss ( ) and time of flight to identify and particles. Events are then selected if they contain exactly one and one . This step can efficiently suppress background events where the final state contains e or μ.

For each photon, the algorithm scans all charged tracks to find the minimum angle between the photon and the tracks. A photon is considered valid only if this minimum angle exceeds 20° and its energy is greater than . This criterion helps reject noise photons produced by hadronic showers. The expected number of final state photons in signal events is from two s. Considering the presence of noise photons in the EMC, the algorithm selects events with at least photons, which suppresses events without in the τ decay.

To further eliminate noise photons, a machine learning approach is employed to distinguish signal photons from noise photons. The variables used in the machine learning analysis include:

● gam_energy: Energy of the EMC photon cluster (GeV).

● gam_secmom: Second moment of the EMC cluster, describing the shower shape (energy spread).

● gam_a42mom: moment of the EMC cluster, a higher-order shower shape variable.

● gam_hits: Number of crystals hit in the cluster.

● gam_var2: , energy concentration in array.

● gam_var3: , average non-seed energy normalized by seed energy.

The distributions for signal and noise photons are shown in Fig. 1. The distinct differences between signal and noise photons enable effective photon selection using machine learning. We train models using boosted decision trees (BDT) and boosted decision trees with gradient boosting (BDTG) with the TMVA toolkit [21], and the resulting receiver operating characteristic (ROC) curves are shown in Fig. 2. BDTG performs slightly better than BDT, and therefore BDTG is used in a subsequent analysis.

The BDTG response distributions for training and testing datasets are shown in the left panel of Fig. 3, which indicate no significant overfitting as the distributions agree quite well. The BDTG cut efficiency is presented in the right panel of Fig. 3. Considering both signal efficiency and signal-to-noise ratio, a BDTG cut of is chosen, leaving approximately 84% of signal photons and 37% of noise photons.

After machine learning selection, the effective number of photons for mode events is typically . Consequently, the analysis algorithm selects events with photons.

Although we can reconstruct the tracks of final-state , , and photons, it is unclear which photons are signal photons and how these photons associate with the corresponding or in the mode. The objective of this step is to determine the best pairing of these particles. Various methods were designed and compared for selecting the optimal pairing, with the best approach determined by the value from a joint kinematic fitting, as detailed in Section III. The joint kinematic fitting not only determines the best pairing but also improves the signal-to-noise ratio by selecting events with .

To further enhance the signal-to-noise ratio, event-level machine learning is applied to select signal events. The variables used in this analysis include the momentum of the , , and the four photons (in sequence). Similar to photon selection using machine learning, the BDTG response and cut efficiency are shown in Fig. 4. A BDTG cut of is selected, which results in about 73% of the signal events and 32% of the remaining background events.

To obtain optimal observables, it is necessary to reconstruct for the momentum of the τ lepton, as detailed in Section IV.A. During this process, scenarios may arise with two solutions, one solution, or no solution. Events with no solution are considered background and excluded.

The selection algorithms applied reduced the initial events to events. The selection efficiency at each step is summarized in Table 1. The overall selection efficiency is 0.49%, and the signal ( ) efficiency is 6.3%, with the signal purity increased from 6.2% to 80.0%.

Detailed event-type analysis on the selected events with a generic topology analysis package, TopoAna [22], shows that background decays have been almost entirely filtered out. The dominant background processes after selection involves more , such as (about 14%), which may require further selection optimization.

Considering that at the CME of , the STCF is expected to produce approximately τ lepton pairs per year [13], of which around are signal events of the mode, we estimate a signal yield of per year after selection, with a signal efficiency of 6.3% and a signal purity of 80.0%. For comparison, up to 2022, Belle experiments have events of the mode. After selection, the signal yield is , with a signal efficiency of 6.3% and a signal purity of 82.4% [11]. The current feasibility study shows that the tau pair selection efficiency is comparable to that of the B factory. After 10 years of operation, the STCF will collect tau pairs of mode after reconstruction, which is orders of magnitude higher than that of Belle. In addition, the event selection at STCF can be further optimized with the vertex detector. This increase in statistics will improve the sensitivity of the τ lepton EDM measurement.

Owing to the presence of neutrinos among the τ lepton decay products, which cannot be detected by the instruments, it is impossible to fully reconstruct the final-state particles to determine the momentum of the τ lepton. Therefore, a special method is required to calculate the τ lepton momentum.

After event selection and particle pairing, the momenta and energies of , and have been obtained from the track information of the main drift chamber and photon signals in the electromagnetic calorimeter. To accurately determine the τ lepton momentum, analytical computation is used to derive the solutions.

In electron-positron collider experiments, the beams are designed to collide at a small angle at an interaction point to optimize collision performance and data collection efficiency. In the laboratory frame, the total momentum of the system is given by , where is a small value that can be determined from the total energy and design parameters of the experiment. By applying a Lorentz transformation to all particles to move into the center-of-mass frame, we have

17

and the energies of the and leptons become

18

The magnitude of the can then be derived from as

19

Next, we solve for the direction of . By combining Eqs. (9), (10), (14), and (17), the cosine of the angle between the vectors and is given by [23]

20

where

21

22

In the center-of-mass frame, we have

23

Given that the angle between the vectors and is , and the angle between the vectors and is , must lie on the intersection of two cones: one with axis and half-angle , and the other with axis and half-angle . Using Mathematica, two analytic solutions for were obtained. There are square root operations in the expressions for the solutions. If the term under the square root is negative (small negative values because of experimental uncertainties are treated as zero in this study), the corresponding case has no physical solution and is therefore discarded. If the term is positive, it yields two distinct solutions. Among these, only one corresponds to the physical reality. Experimentally, this ambiguity cannot be resolved, and therefore, following the same approach as the Belle experiment [11], this study adopts the method of taking the average of the two solutions.

After determining in the center-of-mass frame, a Lorentz transformation is applied to convert it back to the laboratory frame, resulting in the reconstructed momentum of the τ lepton. For signal events, the relative deviation of the reconstructed transverse momentum and angular distribution between the reconstructed and truth τ lepton momentum are shown in Fig. 7. The full width at half maximum of the relative deviation in transverse momentum is , and the peak opening angle is with a root mean square (RMS) of .

The decay products keep the information of the spin of the tau lepton, which can be used to reconstruct the polarimeter vector. For the hadronic decay channel of the tau lepton, we have the decay matrix element as [18]

24

25

26

with

27

28

where represents the momenta of and represents the momenta of the corresponding neutrinos. J will be different for different decay channels

29

30

Combining the decay and production, we obtain for each component of the produciton matrix element,

31

Then, the optimal observable can be constructed using the final state momentum. However, as discussed above, we can only determine the momenta of the neutrino/tau lepton up to a two-fold ambiguity. Then, the matrix element calculated from these two solutions will be averaged when calculating the optimal observable.

MadGraph [24] is used along with custom UFO model files to simulate the production of τ lepton pairs for different values of . Following the reconstruction of the final states in the previous sections, a fast simulation of the detector response is performed with Delphes [25]. The distributions of the optimal observables for several different choices of from the channel are shown in the left panel of Fig. 8. The mean values of the optimal observables are calculated, and a linear fit is performed based on Eq. (7), which is shown in the right panel of Fig. 8. The corresponding result of the linear fit is given in Table 3 with the estimated error on the measurement of the mean value of the optimal observables. The estimated sensitivity after 10 years of operation at the STCF is at a 68% confidence level. This provides a basis for estimating the EDM of the τ lepton in future experiments at STCF.

This method has two steps. First, are paired into , and into , followed by kinematic fitting (using the constraints in Eqs. (11) and (13)). Considering the equivalence of and , there are possible pairings. The algorithm traverses all pairings and selects the one with the smallest sum of from the kinematic fitting. Second, is paired with and with for another kinematic fitting (with the constraints in Eqs. (9), (10), (12), and (14)); the reverse pairings are attempted with the fit yielding the smaller being selected (while still requiring that the neutrino momentum satisfies Eq. (16)). The particle pairing accuracy in this method is lower than that of Method I (joint kinematic fitting) beause of the two-step kinematic fitting process, where separate determinations lead to a sequence of locally optimal choices that may not be globally optimal.

In τ decays ( ), and form a ρ resonance [20], thereby enabling the inclusion of the ρ mass constraint:

A1

Given the broad mass range of the ρ resonance (indicating a large uncertainty), directly adding this constraint to the kinematic fitting would force the fit to adjust physical quantities to strictly satisfy the constraint, which is not desirable. Instead, using the energies and momenta obtained from the kinematic fitting in the second step of Method II, one can calculate the corresponding ρ masses using Eq. (A1), compare them to the theoretical mass, and select the pairing with the smallest deviation. Using this method, the particle pairing accuracy is still lower than that of Method I.

In this method, after performing the kinematic fitting for all possible pairings, the neutrino momenta are obtained, and the τ lepton momenta are calculated as and . For the correct pairing, the angle between the momenta of the final-state particles and the τ lepton momenta (considering only the acute angle) should be smaller. Let denote the unit vector along and

A2

Larger T values are more likely to correspond to the correct physical scenario [26], and therefore, this method selects the pairing with the largest T value as the chosen pairing. Results indicate that this method yields a higher pairing accuracy than those of Methods II and III; however, it is still inferior to that of Method I.