Introduction

Quantum chromodynamics (QCD) predicts a phase transition from hadronic to deconfined partonic matter at sufficiently high temperatures and/or densities [1]. Results from top RHIC and LHC energies indicate that a new form of matter with a low viscosity and high temperature, quark-gluon plasma (QGP), has been produced [2-7]. Lattice QCD calculations predict that the phase transition from hadronic matter to the QGP phase is a smooth crossover in the vanishing baryon chemical potential (μ_B) region [8]. A first-order phase transition is expected in a finite baryon chemical potential region, and thus, determining the phase structure of the QCD is a major research goal in the field of medium- and high-energy heavy-ion collisions [9-12].

The Cooling Storage Ring external-target experiment (CEE) is a spectrometer that is employed to investigate the properties of nuclear matter in the 2.1–2.4 GeV energy region in the center-of-mass frame [13]. The CEE primarily allows near-full-space measurements of charged particles in heavy-ion collisions and provides experimental data for studying important scientific problems, such as spin- and isospin-related nuclear forces, nuclear matter equations of state, and QCD phase structures at high baryon number densities [14-16]. This offers valuable research opportunities for QCD phase diagram studies in low-temperature and high-baryon-density regions.

The event anisotropy of final-state particles relative to the reaction plane in momentum space, also known as collective flow [17], is important for evaluating the properties of media created in heavy-ion collisions. Flow coefficients, such as directed flow v₁ and elliptical flow v₂, are characterized by harmonic coefficients in the Fourier expansion of the azimuthal distribution of the final particles with respect to the reaction plane. In heavy-ion collisions, the driving force of the collective flow is the initial anisotropy in coordinate space. It rapidly diminishes as a function of time, and this phenomenon is known as the self-quenching effect. Thus, the collective flow is sensitive to the details of the expansion of nuclear matter during the early collision stage. The directed flow v₁ was predicted to be sensitive to the effective equation of state [18-20]. The elliptic flow v₂ is sensitive to the constituent interactions and degrees of freedom [21-23]. CEEs can reveal the collective flow in heavy-ion collisions at $\sqrt{s_{{}_{\text{NN}}}}$ = 2.1–2.4 GeV. This will help us study the medium properties and further search for possible QCD phase transition signals [24-26]. One of the principle functions of the zero-degree calorimeter (ZDC), a subdetector of the CEE, is to determine the reaction plane in nuclear–nuclear collisions. The reconstructed reaction plane (usually called the event plane) is crucial for many measurements, such as collective flow [27-29], azimuthal HBT [30], and CME-related observables [31-34].

In this paper, we introduce the necessary acceptance corrections and calibrations for event-plane determination from the CEE-ZDC. Furthermore, the isospin-dependent quantum molecular dynamics (IQMD) model [35] is used to predict collective flow from a typical CEE energy ($\sqrt{s_{{}_{\text{NN}}}}$ = 2.1 GeV).

CEE-ZDC

Figure 1(a) shows a schematic of the CEE spectrometer. The detector subsystem consists of a superconducting dipole magnet used to deflect charged particles; a silicon pixel positioning detector (SiPiX, Beam Monitor) to measure the position and time of the incident beam as well as the primary collision vertex [36]; A time projection chamber (TPC) is used to reconstruct particle trajectories and identify particles [37], and a time-of-flight chamber (TOF) is employed to extend particle identification to high momentums (p > 2 GeV/c) The TOF chamber contains a start-time detector (T0) [38], an inner time-of-flight detector (iTOF) [39], and an end-cap time-of-flight detector (eTOF) [40]. In addition, multi-wire drift chambers (MWDCs) are designed to track charged particles at forward rapidity and identify particles via momentum measurements [41]. The ZDC is used to measure the patterns (deposited energy and incident position) of forward-going charged particles emitted from nuclear–nuclear collisions [42].

Fig. 1Full-screen View

(Color online) (a) Schematic of the CEE detector. (b) Schematic of the ZDC detector.

The ZDC is proposed to be installed behind all the other subdetectors. The beam direction is defined as the positive Zaxis, and the ZDC is located at Z = 295–299 cm, facing the original incident beam direction; iIts geometry is shown in Fig. 1(b). The ZDC detector cross-plane is a wheel with radius R ranging from 5 to 100 cm, and the vacuum pipe carrying the nuclear beam passes through the inner hole of this ZDC wheel. It consists of 24 sectors that subtend an azimuthal angle of 15. Each sector is divided into eight modules forming eight rings in the full ZDC plane. The sensitive volume of the ZDC is composed of a plastic scintillator, and the current design uses the BC-408 material from Saint-Gobain [43]. The photons are produced inside the scintillator through the deposited energy of the incident particles and are then transported through a plastic light guide into the quartz window of a traditional PMT. ZDC covers the pseudo-rapidity range between 1.8 and 4.8, allowing the determination of the centrality and event plane in the forward rapidity region and minimizing autocorrelations from middle rapidity analyses [17, 44].

Event plane determination from the CEE-ZDC

In the study of the event plane, the simulation input of ²³⁸U + ²³⁸U collisions at 500 MeV/u was obtained from the IQMD generator [35]. The IQMD model was developed based on the quantum molecular dynamics (QMD) model [45] considering isospin effects. The detector environment was simulated using GEANT4 [46]. One million IQMD simulated events were generated in the range of the nuclear impact parameter b, which is the transverse distance of the projectile from the target nucleus, 0 < b < 10 fm, with 0.1 million events for each b interval of 1 fm.

The reaction plane in the nucleus-nucleus collision is defined by the vector of the impact parameter and beam direction. As the impact parameter could not be directly measured in the experiment, the reaction plane was estimated using the standard event plane method [47, 17]. The first-order harmonic event plane Ψ₁ is calculated using the event flow vector Q₁: ${\text{Q}}_1=\left(\begin{array}{c}{\displaystyle {\sum }_i{w}_i}\sin \left({\varphi }_i\right)\\ {\displaystyle {\sum }_i{w}_i}\cos \left({\varphi }_i\right)\end{array}\right)\text{, }{\text{Ψ}}_1={{\displaystyle \tan }}^{-1}\left(\frac{{\displaystyle {\sum }_i{w}_i}\sin \left({\varphi }_i\right)}{{\displaystyle {\sum }_i{w}_i}\cos \left({\varphi }_i\right)}\right),$ (1) where the sum exceeds all particles used in the flow vector calculation. Quantities ϕi are the azimuths in the laboratory frame. The weight wi is defined by the deposited energy Δ E of particle i collected by ZDC detector. As it is related to the mass and transverse momentum p_T of the particle, whereas the p_T weight is commonly applied in flow analysis to optimize the event plane resolution [47]. The smearing effect of the deposited energy is considered by Equ. 2 $\begin{array}{l}\Delta E=\Delta E^{\prime }\times \left[1-\frac{1}{4}{\left(\frac{L}{5.5}\right)}^2\right]\mathrm{,}h<8\\ \Delta E=\Delta E^{\prime }\times \left[1-\frac{1}{4}{\left(\frac{L}{5.5}\right)}^2\right]\times \left[8+\frac{2}{3}\left(h-8\right)\right]\mathrm{,}h\ge 8\end{array}$ (2) where L is the distance from the hit position to the geometric center of the sector. h is the charge of the final particle. The term $1-\frac{1}{4}{\left(\frac{L}{5.5}\right)}^2$ is used to describe the resolution of the deposited energy at the edge of the sector. and the term $8+\frac{2}{3}\left(h-8\right)$ is used to simulate the saturation effect of the deposited energy resolution for heavy nuclei (h≥8) [48].

Because the finite multiplicity limits the estimation of the reaction plane, it yields a resolution factor R which is defined by Equ. 3. In this study, we focus on the first-order harmonic event plane because v₁ is more significant than higher-order flows in the collision energy range covered by CEE. $R_1=\langle \cos \left({\text{Ψ}}_{\mathrm{1,}\text{EP}}-{\text{Ψ}}_{\mathrm{1,}\text{RP}}\right)\rangle$ (3) The magnetic field direction was perpendicular to the beam direction at CEE. Thus, the charged particles in the final state are deflected by the magnetic field and hit one side of the ZDC detector more, as shown in Fig. 2(a) Owing to the asymmetric ZDC acceptance, The reconstructed event-plane angle was not isotropic within the laboratory frame. but is biased towards the π-azimuth. The acceptance bias caused by the magnetic field introduces additional nonphysical anisotropy for the detected collision events, and this effect should be removed as it distorts the event plane reconstruction. Therefore, we introduce a position weight to calibrate asymmetric acceptance.

Fig. 2Full-screen View

(Color online) (a) Hit distribution obtained from the ZDC with a collision impact parameter of 5 < b < 6 fm. (b) Hit distribution obtained from the ZDC after position weight correction with a collision impact parameter of 5 < b < 6 fm.

The core idea of the position weight is to correct the asymmetric acceptance of the ZDC caused by the magnetic field. Owing to the deflection of charged particles in the magnetic field, The left side of the ZDC detector receives more hits. We assigned a weight P of less than 1 to the hits on the left side to correct this effect. The weight was calculated based on a two-dimensional X-Y hit distribution, as defined in Equ. 4 is the ratio of the number of hits on the right side to those on the left side. In addition, the deposited energy Δ E was used as the weight when calculating the number of hits. because it is related to particle mass. It can be observed that the acceptance of ZDC is symmetric after applying the position weight, as shown in Fig. 2(b). $\begin{array}{l}{w}_i=\Delta E\times P\\ P=n\left(-x\mathrm{,}y\mathrm{,}\Delta E\right)/n\left(x\mathrm{,}y\mathrm{,}\Delta E\right)\mathrm{,}x<0\\ P=\mathrm{1,}x>0\end{array}$ (4)

The black line in Fig. 3 shows the event plane distribution before position weight correction. With an ideal detector, the event plane distribution should be flat because the possible direction of the impact parameter b is random in the 2π azimuths of the transverse plane in the laboratory frame. It is not flat but peaks around ${\text{Ψ}}_1\sim \pi$ owing to the asymmetric acceptance of ZDC as discussed above. Correspondingly, one can see that the resolution difference between the left (azimuth of the reaction plane: π/2 to 3π/2) and right sides (-π/2 to π/2) of ZDC is significant, as shown in Fig. 4(a). After applying the position weights defined in Equ. 4: The unflatness of the event plane is significantly reduced, as indicated by the red line in Fig. 3. The resolution difference between the left and right sides of ZDC is significantly reduced, as shown in Fig. 4(b). This indicates that the position weight naturally corrected the acceptance asymmetry of ZDC.

Fig. 3Full-screen View

(Color online) Event plane distributions before position weight correction (black line), after position weight correction (red line), and after position weight + shift corrections (blue line).

Fig. 4Full-screen View

(Color online) (a) Resolution of 1st order event plane as a function of impact parameter b without position weight. (b) Resolution of 1st order event plane as a function of b with position weight. (c) Resolution of 1st order event plane as a function of b with position weight and shift correction.

The event plane distribution is not perfectly flat after the position weight, as shown in Fig. 3. Consequently, the resolution difference between the left and right sides of ZDC was still visible. Therefore, the shift method is used to force the event plane to be flat [47]. A shift angle Δ Ψ₁ is applied to correct the event plane. and Δ Ψ₁ is calculated event-by-event using the following equation: $\begin{array}{l}{{\text{Ψ}}^{\prime }}_1={\text{Ψ}}_1+\Delta {\text{Ψ}}_1\\ \Delta {\text{Ψ}}_1={\displaystyle \sum _{i=1}^{20}\frac{2}{i}}\left[-\langle \sin \left(i{\text{Ψ}}_1\right)\rangle \cos \left(i{\text{Ψ}}_1\right)+\langle \cos \left(i{\text{Ψ}}_1\right)\rangle \sin \left(i{\text{Ψ}}_1\right)\right],\end{array}$ (5) where the brackets refer to the average over the events in the same centrality bins. Ψ₁ is the position-weight-corrected event plane azimuth, and ${{\text{Ψ}}^{\prime }}_1$ is the event plane angle with shift calibration. After the shift calibration, a flat event plane distribution was achieved, as indicated by the blue line in Fig. 3. The resolution between the left and right sides was consistent, as shown in Fig. 4(c).

In the experiment, the event plane calculated from different rapidity windows helped us understand the systematic uncertainties in the flow measurements. Correspondingly, the event planes from ZDC sub-rings, which correspond to different rapidity windows, were studied. Figure 5 shows the 1st order event plane resolution from ZDC sub-ring radius 52.5 < R < 76.25 cm without a position weight Fig. 5a, with a position weight Fig. 5b, and with a position weight and shift correction Fig. 5c. These results indicate that the position weight and shift methods work well for the event plane calculated by ZDC subring.

Fig. 5Full-screen View

(Color online) (a) Resolution of 1st order event plane, determined from the ZDC sub-ring (radius: 52.5 < R < 76.25 cm) without position weight. (b) Resolution of 1st order event plane, determined from the ZDC sub-ring (radius: 52.5 < R < 76.25 cm) with position weight. (c) Resolution of 1st order event plane, determined from the ZDC sub-ring (radius: 52.5 < R < 76.25 cm) with position weight and shift corrections.

After eliminating the resolution difference due to asymmetric acceptance using the position weight and shift methods, the 1st-order event plane resolution from the ZDC was calculated using the two-sub-event plane method [17]. The full event was divided randomly into two independent sub-events with equal tracks, and the event-plane resolution was estimated by correlating the two sub-events. as defined by Eq. 6: $\begin{array}{c}{R}_{\mathrm{1,}\text{sub}}=\sqrt{\langle \cos \left({\text{Ψ}}_1^A-{\text{Ψ}}_1^B\right)\rangle }\\ =\sqrt{\pi }/2\chi e^{(-{\chi }^2/2)}\left(I_0\left({\chi }^2/2\right)+I_1\left({\chi }^2/2\right)\right),\end{array}$ (6) where A and B denote the two subevents. As χ is proportional to the square root of the multiplicity and a full event with two particles as subevents, The full event-plane resolution is obtained as follows: $R_{\text{full}}=R\left(\sqrt{2}{\chi }_{\text{sub}}\right)$ (7) The resolution of 1st-order event plane as a function of impact parameter, determined from the ZDC whole ring, is compared with the 1st-order event plane resolution, determined from the STAR event plane detector, for Au+Au collisions at $\sqrt{s_{{}_{\text{NN}}}}$ = 3.0 GeV [28] in Fig. 6. The event plane resolution from CEE-ZDC reached ~ 90% in the middle-central collisions (4 < b < 7 fm). The resolution of ZDC is better in the region of b < 4 fm but worse for b > 4 fm, which is probably due to the different sizes of the gold and uranium nuclei, experimental acceptance, and detector performance.

Fig. 6Full-screen View

(Color online) Resolution of 1st order event plane, determined from the ZDC whole ring, as a function of impact parameter compared with that of 1st order event plane determined from STAR event plane detector for Au+Au collisions at $\sqrt{s_{{}_{\text{NN}}}}$ = 3.0 GeV.

We also systematically investigated the effects of the ZDC detector thickness, hit efficiency, energy resolution, and model dependence on the first-order event plane resolution. As shown in Fig. 7, where the solid red dots represent the default conditions: a ZDC thickness of 4 cm, hit efficiency of 100%, and default energy smearing, as shown in Eq. 2 and heavy nuclei from IQMD generator de-excitation. The effects of the different variables were investigated individually. The resolution of 1st-order event plane decreases slightly as the ZDC detector thickness decreases, as shown in Fig. 7(a). This is because a more accurate measurement of the deposited energy is achieved with a thicker ZDC. Fig. 7(b) shows the hit efficiency dependence of 1st-order event-plane resolution. The ZDC hit efficiency was reduced to 90%, and the event-plane resolution remained almost unchanged. The effect of ZDC energy resolution is investigated by applying additional Gaussian smearing to the deposited energy, where Gaussian(1, 0.5) has a center value of 1 and width of 0.5, and Gaussian(1, 1) has a center value of 1 and width of 1. A smaller Gaussian width indicates a better energy resolution. As the energy resolution decreases, the first-order event-plane resolution decreases by approximately 5-10%, as shown in Fig. 7(c). Figure 7(d) shows the relationship between the ZDC event plane resolution and the IQMD heavy nuclei de-excitation, where "out"/"in" means the heavy nuclei are de-excitation or not. The resolution estimated using the IQMD model with heavy nuclei de-excitation was slightly higher than IQMD without heavy nuclei de-excitation because the multiplicity was higher in the former case.

Fig. 7Full-screen View

(Color online) (a) Effect of ZDC thickness on the 1st order event plane resolution. (b)Effect of ZDC hit efficiency on the 1st order event plane resolution. (c)Effect of ZDC energy resolution on the 1st order event plane resolution. (d)Effect of heavy nuclei de-excitation on the 1st order event plane resolution.

Collectivity flow predictions from IQMD model

Collective flow is sensitive to the details of the expansion of the medium produced during the early collision stage. Flow measurements at CEE provide information on the QCD phase structure in the high-baryon-density region. Collectivity flow predictions were presented for a typical CEE based on IQMD model. Figure 8 shows v₁ and v₂ as functions of rapidity for protons, deutons, tritons, ³He, and ⁴He with an impact parameter of 1 < b < 4 fm from IQMD ²³⁸U + ²³⁸U collisions at 500 MeV/u ($\sqrt{s_{{}_{\text{NN}}}}$ = 2.1 GeV). The v₁ slope values extracted using y=ax+bx³ strongly depend on the number of nuclei. The v₂ values are negative in the middle rapidity owing to the squeeze-out effect; the medium expansion is shadowed by spectator nucleons, and particles are preferentially emitted in the direction perpendicular to the reaction plane [17], whereas v₂ becomes positive in the forward rapidity as the squeeze-out effect becomes weak. Similar to v₁ slope, v₂ values showed a strong dependence on the number of nuclei.

Fig. 8Full-screen View

(Color online) v₁ and v₂ as functions of rapidity, for protons, deuterons, tritons, ³He, and ⁴He with a collision impact parameter 1< b < 4 fm, derived from IQMD ²³⁸U + ²³⁸U 500 MeV/u ($\sqrt{s_{{}_{\text{NN}}}}$ = 2.1 GeV). The v₁ slopes are extracted using the equation: y=ax+bx³.

Figure 9 presents the $d\left(v_1/A\right)/dy$ and v₂ for protons, deuterons, tritons, ³He, and ⁴He determined from HADES [49] and STAR [50] experiments together with the IQMD model calculations¹, where A denotes the atomic number. v₁/A represents the directed flow carried by each nucleon in the light nuclei, and the scaling behavior suggests a coalescence production mechanism of the light nuclei during heavy-ion collisions. v₂ is calculated in the rapidity range of -0.1 < y < 0 for the STAR experiment and IQMD model calculations and -0.05 < y < 0.05 for the HADES experiment. The atomic-number-scaled v₁ slope from HADES and IQMD showed a decreasing trend with an increase in atomic number, whereas the STAR data weakly depend on the atomic number in the collisions at $\sqrt{s_{{}_{\text{NN}}}}$ = 3 GeV. The absolute value of v₂ from HADES decreased with increasing atomic number, whereas the results for STAR and IQMD remained almost unchanged with atomic number. This may indicate that light nuclei are not purely formed by the coalescence mechanism in Au+Au collisions at $\sqrt{s_{{}_{\text{NN}}}}$ = 2.4 GeV, Coalescence was the dominant production mechanism for Au+Au $\sqrt{s_{{}_{\text{NN}}}}$ = 3.0 GeV. The production of light nuclei in the IQMD model is a mixture of light nuclei fragments and the coalescence of nucleons and light nuclei. The dominant production mechanism in the IQMD model depends on the collision energy and parameter settings. The predictions provided by the IQMD model in U + U collisions at $\sqrt{s_{{}_{\text{NN}}}}$ = 2.1 GeV will be validated in future CEEs.

Fig. 9Full-screen View

(Color online) Atomic number A scaled v₁ slope (upper panel) and v₂ (lower panel) at middle rapidity for protons, deuterons, tritons, ³He, and ⁴He determined from HADES, STAR experiments, and IQMD model calculations for CEE. The brackets represent the systematic uncertainties in the experimental data.

Future measurements of v₁ and v₂ will help us to examine the equation of state of the produced nuclear matter at CEE energies [51, 52] and understand the production mechanism of light nuclei in the high baryon density region [15, 53-56].

Summary

In this paper, we elucidate the procedures for event plane determination from the ZDC at the CEE. The calculated values determined using the IQMD Monte Carlo event generator (500 MeV/u ²³⁸U + ²³⁸U) were incorporated as inputs, and the detector environment was simulated using GEANT4.

To correct for the bias caused by the dipole magnet, a position-dependent weight was introduced to calibrate the asymmetric acceptance. After an additional shift correction, an outstanding first-order event plane resolution of ~ 90% was obtained for middle-central collisions (4 < b < 7 fm). Herein, the collective flows v₁ and v₂, as functions of rapidity for p, d, t, ³He, and ⁴He in middle central collisions, are presented based on the IQMD model. These results were compared with the experimental data obtained from 2.4 GeV and 3 GeV Au+Au collisions in the HADES and STAR experiments, respectively.

The measurements from the HADES and STAR experiments suggest that coalescence is the dominant production mechanism of light nuclei at 3 GeV, whereas light nuclei fragmentation and coalescence are both important at 2.4 GeV. The predictions of the IQMD model at 2.1 GeV will be validated in future CEEs.