Introduction

Quantum chromodynamics (QCD) predicts a transition from hadronic matter to deconfined quark–gluon matter at sufficiently high temperature and/or high density [1, 2]. Heavy-ion collision experiments at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) have provided unique experimental evidence for this transition [3-8]. While striking progress has been made in the past decades, some foundational questions remain to be determined, such as the existence of a critical end point in the QCD phase diagram and the equation of state of nuclear matter at baryon densities much larger than the saturation density [9, 10]. At HIRFL-CSR energies, a hadronic gas with densities reaching 2–3 times nuclear saturation density, where temperatures around 40 MeV can be produced [11]. Experiments at these energies are vital to elucidate the properties of QCD in the low-temperature and high-baryon-density region [12-15].

The CEE spectrometer is designed to measure charged particles in the fixed target heavy-ion collisions at HIRFL-CSR [16]. It is the first comprehensive nuclear physics experiment research facility in the GeV energy range in China [11]. With the various types of ion beam provided by HIRFL-CSR, e.g. ¹²C + ¹²C at 1.1 GeV/u ( GeV) and ²³⁸U + ²³⁸U at 500 MeV/u ( GeV) [17, 18], CEE spectrometer is an ideal platform to explore the QCD phase diagram in the high baryon density region, to study the nuclear matter equation of state and to search for the existence of the critical end point.

Collective flow is one of the most important observables in relativistic heavy-ion collision experiments for studying the bulk behavior of the created matter [19]. The azimuthal anisotropy of emitted particles in the momentum space can be expanded in a Fourier series [20]:(1)where Ψ_RP is the azimuthal angle of reaction plane defined by the beam direction and impact parameter. The Fourier coefficients is the n^th-order flow coefficient, and the bracket means the average over all particles and events. The first harmonic (v₁) is referred as directed flow, found to be sensitive to the compressibility of dense matter [21, 22].

The reaction plane angle is not directly measurable in the heavy-ion collision experiment, but one can use the observed event plane angle Ψ_EP from the anisotropic flow itself as an estimate on an event-by-event basis. This is the standard event plane method [23]. This approach has been widely applied in collective flow analyses over the past decades [24-31], yet it remains crucial to optimize it for the CEE experiment and to ensure a reliable collective flow signal.

This paper is organized as follows. Section 2 describes the setup of CEE detector and the simulation tools. Section 3 presents the reconstruction and correction method of event plane from different sub-detectors of CEE experiment. Section 4 discusses the results of proton v₁ simulated ²³⁸U + ²³⁸U collisions at GeV. A summary is presented in Sec. 5.

CEE Detector and Fast Simulation

The CEE spectrometer is designed for charged particle reconstruction in HIRFL-CSR energy region [12, 32]. Figure 1 presents the primary detector configurations. The main components of CEE spectrometer are as follows: a large-gap dipole magnet with a 0.5 T magnetic field along the y-axis [33, 34]; a Time Projection Chamber (TPC) [35, 36] system consisting of two identical TPCs surrounded by the inner Time-of-Flight (iToF) [37, 38] detectors located in the mid-rapidity region; three layers of Multi Wire Drift Chamber (MWDC) [39-42] followed by the external Time-of-Flight (eToF) [43, 44] detectors and Zero Degree Calorimeter (ZDC) [45, 46] in the forward region; the start time (T0) [47, 48] detector and a silicon pixel beam monitor (BM) [49, 50], which are installed on the beam line in the upstream side of the target to track the beam position. The target is placed at (0, 0, -35) cm to maximize the pion acceptance for ²³⁸U + ²³⁸U collisions at GeV, where the (0, 0, 0) cm is set to the center of TPC geometry. The magnetic field from the dipole magnet is along the y-axis, causing the beam and final-state charged particles to bend along the x-axis. Detectors in the forward region, such as eToF and ZDC, are shifted along the x-axis accordingly to allow the beam to pass through their hollow centers, as shown in Fig. 1. In this paper, the JAM model [51, 52] is used to generate simulated events of ²³⁸U + ²³⁸U collisions at GeV followed by CFS package to simulate the CEE experiment.

Fig. 1Full-screen View

(Color online) The sketch of CEE detector [16]

In JAM, the initial position of nucleon is sampled according to the distribution of nuclear density. All hadronic states, including resonances, are propagated in space-time with explicit trajectories. Inelastic hadron-hadron collisions are described using two approaches: at low energies, resonance production dominates, while at high energies, the color string picture becomes the primary mechanism. The model includes two modes: cascade and mean-field. In the cascade mode, hadrons and their excited states follow straight trajectories in two-body collisions. The nuclear mean-field mode incorporates the interactions of hadrons with the nuclear medium and the equation of state, which is implemented based on the relativistic quantum molecular dynamics approach (RQMD) [53, 54].

The final-state particles generated by the JAM model are processed through the CEE Fast Simulation (CFS) framework. This framework simulates the CEE detector environment and produces responses for all CEE sub-detectors. The CFS enhances computational efficiency through parametric modeling and analytically derived formulations, which collectively simulate critical sub-detector characteristics such as detector acceptance, momentum resolution, energy deposition, and particle flight time. Each sub-system’s resolution effects are implemented via Gaussian smearing of the true input values. This methodology systematically accounts for measurement uncertainties while maintaining an optimal computational efficiency.

Figure 2 shows the proton acceptance of TPC, MWDC and ZDC for events with impact parameter 5 fm < b < 6 fm from JAM + CFS simulation. The top panel of Fig. 2 shows the angular coverage of TPC (Fig. 2(a)) and MWDC (Fig. 2(b)) and the spatial coverage of ZDC (Fig. 2(c)). The CEE spectrometer covers polar angles from 10 ° to 120 ° in the laboratory frame, corresponding to proton rapidities between -0.7 and 1 in the center-of-mass frame. A clear efficiency loss in TPC azimuth at 90 ° and 270 ° is shown in Fig. 2(a), which is due to the two-half design of TPC [36]. Figure 2(c) shows the two-dimensional X-Y hit distribution on ZDC. Clearly, the left side (X < 0) of ZDC receives more hits than the right side (). This is because the final state charged particles are deflected by the magnetic field (along the y-axis), making them more likely to hit one side of ZDC [55]. The bottom panel of Fig. 2 shows the kinematic coverage of TPC, MWDC and ZDC. The dashed box in Fig. 2(d) indicates the kinematic range (0.2 GeV/c < p_T < 0.7 GeV/c and -0.5 < y < 0.5) used for the directed flow simulation of charged particles measured by TPC. Further details are discussed in Sect. 4.

Fig. 2Full-screen View

(Color online) Simulated proton acceptance from CFS package for 2.1 GeV ²³⁸U + ²³⁸U collisions with 5 fm < b <6 fm. Top panel: proton track/hit distribution in TPC (a), MWDC (b) and ZDC (c); Bottom panel: proton kinematic acceptance in TPC (d), MWDC (e) and ZDC (f). The dashed rectangle in panel (d) indicates the kinematic region used for the proton v₁ analysis

Event Plane Reconstruction with MWDC and ZDC

The reaction plane in heavy-ion collisions is defined by the beam direction and impact parameter, which is not directly measurable. In the experiment, one uses the azimuthal emission angles of detected particles to determine the event plane [23], which is used to estimate the reaction plane. The nth-order event plane angle, Ψ_n, is calculated by the n^th-order flow vector Qn. In this study, we focus on simulating v₁, because its magnitude is significantly larger than that of the higher-order flow coefficients at CEE energy. The 1st-order flow vector Q₁ and the event plane angle Ψ₁ are defined as(2)where the sum is over all particles used in the event plane determination and ωi are weights to optimize the event plane resolution.

The main detectors used for event plane reconstruction in the CEE experiment are MWDC and ZDC. Since the MWDC is a track-based detector and the ZDC is a hit-based detector, the information used to obtain the Q₁ and the correction procedure are different.

For the track-based MWDC, the ϕi used in Eq. 2 denotes the azimuthal angle of the i-th particle (obtained from the particle’s momentum) in the event plane determination and ωi is p_T of the i-th particle. The reaction plane distribution should be isotropic. Due to the finite detector efficiency and acceptance, the detected particles are azimuthally anisotropic in the laboratory system which leads to an anisotropic distribution of the reconstructed event plane [19, 23]. The black line in Fig. 3(a) presents the raw Ψ₁ distribution observed from the MWDC.

Fig. 3Full-screen View

(Color online) (a) 1st-order event plane distribution reconstructed by MWDC for raw distribution (black line), after re-centering correction (red line) and after re-centering + shift correction (blue line); (b) 1st-order event plane distribution reconstructed by ZDC for raw distribution (black line), after position weight correction (red line) and after position weight + shift correction (blue line)

To correct the effect of anisotropic Ψ₁ distribution, the re-centering correction is applied [56]. The first-order flow vector Q₁ is recalculated by subtracting the (Q_x,1, Q_y,1) values averaged over all events, as described by(3)where the q_rec is obtained by averaging over all particles used in event plane determination from all events.

The re-centered Ψ₁ distribution from MWDC is not perfectly uniform as shown by the red line in Fig. 3(a). The remaining anisotropic structure is corrected by shift procedure [57]. For each event, a shift angle ΔΨ₁ is calculated from the following equation:(4)where n is the maximum correction order and the brackets refer to the average over the events used in event plane reconstruction. Ψ₁ is the event plane angle after re-centering correction and is the event plane angle after the shift correction. As indicated by the blue line in Fig. 3(a), a isotropic event plane distribution is obtained after the shift correction.

For the hit-based ZDC, the ϕi used in Eq. 2 denotes the azimuthal angle in the laboratory frame of the i-th particle hit on the ZDC and ωi is the energy deposition ΔE of the i-th particle in the ZDC [55]. The magnetic field direction of the CEE experiment is along the y-axis which is perpendicular to the beam direction. The final state charged particles are deflected by the magnetic field, therefore, more likely to hit on one side of the ZDC, as shown in Fig. 2(c). This acceptance asymmetry of the ZDC will bias the reconstructed event plane toward a π azimuth, as shown by the black line in Fig. 3(b). To correct for this acceptance bias caused by the magnetic field, Ref. [55] proposed a position weight correction to calibrate the asymmetric acceptance as defined in Eq. 5:(5)where the weight wi is defined as the deposited energy ΔE of the i-th particle hit on the ZDC, multiplied by a position-weight factor derived from the two-dimensional X-Y hit distribution. This position weight is calculated by the ratio of the number of hits on the right side of ZDC to that on the left, or to those on the left, using ΔE as the weighting factor [54]. A shift correction is also applied after the position weight correction because the event plane distribution is not perfectly uniform, shown by the red line in Fig. 3(b). The shift angle is calculated from Eq. 4, where Ψ₁ is the position-weight-corrected event plane angle, and is the event plane angle after shift calibration. The resulting distribution, achieved after applying all corrections, is shown by the blue line in Fig. 3(b). Due to the finite multiplicity of final-state particles, the reconstructed event plane deviates from the true reaction plane. We correct for this deviation using the first-order event plane resolution defined in Eq. 6:(6)Since the Ψ_RP is not directly measurable, the first-order event plane resolution from MWDC and ZDC are extracted with two-sub-event plane method [19, 23]. In this approach, each event is randomly divided into two sub-events with equal tracks (MWDC) or hits (ZDC). The event plane resolution of the two-sub-event is calculated by(7)where and are the corrected event plane angle of of two sub-events. Then the full event plane resolution is calculated from Eq. 8:(8)where I₀ and I₁ are the modified Bessel functions, and which is proportional to the square root of event multiplicity. Thus, the full event plane resolution is obtained by [23, 55].

Figure 4 presents the first-order event plane resolution as a function of impact parameter for the MWDC and ZDC, obtained using the two-sub-event method. The resolution from the MWDC is generally higher than that from the ZDC. Both detectors can achieve a maximum resolution (~90% for MWDC and ~70% for ZDC) in the mid-central collisions (5 fm < b < 7 fm). While the absolute resolution values can vary with different input models, the MWDC consistently shows a better first-order event plane resolution than the ZDC. An important observation is the lack of MWDC event plane resolution data for the most central collisions (0 fm < b < 1 fm). This absence is primarily due to significant non-flow effects, which induce a negative correlation between the two MWDC sub-events. Although this effect can generally be corrected in simulation, such a correction is beyond the scope of this paper.

Fig. 4Full-screen View

(Color online) 1st-order event plane resolution as a function of impact parameter b from MWDC and ZDC

Directed Flow Simulation from TPC

With the corrected event plane from MWDC (ZDC) and the corresponding event plane resolution, the directed flow of charged particles detected by TPC is calculated with Eq. 9:(9)where is the azimuth angle of the particle of interest (POI) in the TPC and the bracket denotes the average of all POIs within selected kinematic range in all events with the same event category. In this study, we select the events with impact parameter 5 fm < b < 6 fm for illustration and protons (POIs) within 0.2 GeV/c < p_T < 0.7 GeV/c and -0.5 < y < 0.5, indicated by the dashed box in Fig. 2(d).

Figure 5 compares the proton v₁ calculated using the true reaction plane versus the event planes reconstructed by different detectors. We first validated the CEE detector’s ability to measure the proton v₁ signal by comparing the flow calculated with the true reaction plane () to the theoretical expectation from the JAM model (). Their close agreement, shown in Fig. 5(a), confirms the detector’s fundamental reliability. We then evaluated v₁ using the standard event plane method from both the MWDC () and ZDC (). As shown in Figs. 5(b) and (c), after applying corrections for self-correlation and momentum conservation [19, 23, 58], both and are consistent with . This final result demonstrates that the standard event plane method is valid and applicable for directed flow measurements in the CEE experiment.

Fig. 5Full-screen View

(Color online) Proton v₁ as a function of rapidity extracted from different detectors: (a) proton v₁ from JAM model (gray cross) and v₁ extracted from reaction plane from CFS package (magenta open diamonds); (b) un-corrected proton v₁ (gray filled circles) and corrected proton v₁ (red open circles, corrected for self-correlation and momentum conservation effect) from 1st-order MWDC event plane; (c) un-corrected proton v₁ (gray filled squares) and corrected proton v₁ (blue open squares, corrected for self-correlation) from 1st-order ZDC event plane

It is worth noting that the removal of self-correlation and momentum conservation effect is on a track-by-track basis [19, 23, 58], which is achievable by carefully matching reconstructed tracks from MWDC and TPC. But for ZDC, given the complicated magnetic field and the position of ZDC, it is difficult to do a precise matching between ZDC hits and TPC/MWDC tracks. Thus, it is complicated to remove such effects for . To avoid self-correlation and momentum conservation effect, we propose using the ZDC event plane () exclusively for backward protons (-0.5 < y < 0) detected in the TPC. The results of this proposed measurement are shown in Fig. 6. The dashed lines are fits to extract v₁ slope and the fit function is . The dv₁/dy is 0.631 ± 0.002 for , 0.629 ± 0.003 for and 0.634 ± 0.012 for , respectively. The v₁ slopes extracted from and are consistent with that from within 1σ, which means the proposed measurement is reasonable.

Fig. 6Full-screen View

(Color online) Comparison of proton v₁ extracted from reaction plane (magenta open diamonds), 1st-order MWDC event plane (red open circles) and 1st-order ZDC event plane (blue open squares). The dashed lines are fits to proton v₁ extracted from corresponding planes. The fit function is

Furthermore, to study the influence of detector effect on the flow measurement, detector efficiency is introduced into the CFS package. A 90% detection efficiency is applied to the TPC, MWDC, and ZDC. Figure 7 displays the results for 100% (symbols) and 90% (bands) for clarity. The consistency between v₁ obtained with 100% and 90% efficiency indicate that the influence of detector efficiency is negligible for flow measurement. It should be noted that a complete study would require applying a realistic, detector-specific efficiency; however, this is not expected to alter the general conclusion.

Fig. 7Full-screen View

(Color online) Comparison of proton v₁ with different detector efficiency. Open symbols are proton v₁ extracted with 100% efficiency from MWDC (red open circles) and ZDC (blue open squares). Dashed lines are proton v₁ extracted with 90% efficiency from MWDC (red dashed line) and ZDC (blue dashed line)

Summary

In summary, we have presented a procedure for simulating directed flow using the standard event plane method in the CEE experiment. The JAM model (500 MeV/u ²³⁸U+²³⁸U) is used as input for the simulation, and the CFS package is applied to incorporate the CEE detector environment. The directed flow signal is extracted by correlating charged particles detected in the TPC with the first-order event plane reconstructed by the MWDC and ZDC. The paper also discusses the corresponding correction procedures for both the event plane and the resulting v₁. The consistency among , , and indicated validity of standard event plane method in the CEE experiment. We also propose an optimal kinematic region for v₁ measurement using 1st-order event plane reconstructed by MWDC and ZDC. The framework developed in this work can serve as a guideline for directed flow measurements in future CEE experiments.