Λ reconstruction

Λ particles were reconstructed using the decay channel $\Lambda \to p+{\pi }^{-}$, which offers a branching ratio of 69.2% [4]. Λ particles decayed with an appropriate decay length of cτ ≈ 79 mm after they were produced in Au+Au collisions. Protons and pions were identified based on the ionization energy loss in the TPC gas. In practice, charged tracks with $\left|n{\sigma }_{\text{X}}\right|<3$ for any particle of interest X were selected, where $n{\sigma }_{\text{X}}$ is defined as follows. $n{\sigma }_{\text{X}}=\frac{1}{{\sigma }_{\text{X}}}\log \frac{{\langle \text{d}E/\text{d}x\rangle }_{\text{measured}}}{{\langle \text{d}E/\text{d}x\rangle }_{\text{X}}^{\text{Bischel}}}\mathrm{,}$ (2) where ${\langle \text{d}E/\text{d}x\rangle }_{\text{measured}}$ denotes the average energy loss per unit length measured by the TPC of the STAR detector, ${\langle \text{d}E/\text{d}x\rangle }_{\text{X}}^{\text{Bischel}}$ represents the expected energy loss $\langle \text{d}E/\text{d}x\rangle$ for a certain particle species X (in this case, protons or pions), and ${\sigma }_{\text{particle}}$ denotes the $\langle \text{d}E/\text{d}x\rangle$ resolution measured by the TPC (typically ≈ 8% [9]). For each proton or pion track, we required a minimum of 15 hits in the TPC to ensure adequate track quality.

Using the data collected in the STAR experiment from Au+Au collisions at $\sqrt{s_{\text{NN}}}=27$ GeV, Λ particles were reconstructed using the KF Particle method, and various kinematic and topological variables such as mass, pT, and decay length were calculated. As depicted in Fig. 3, clear Λ mass peaks were observed in the invariant mass $m_{p{\pi }^{-}}$ distributions in the pT range of 0.4–0.6 GeV/c from collision events with 0–5% (left panel) and 30–40% (right panel) centrality.

Fig. 3Full-screen View

$p{\pi }^{-}$ invariant mass distributions of p_T=0.4-0.6 GeV/c in Au+Au collisions at $\sqrt{s_{\text{N}N}}=27$ GeV with centrality 0-5% (left) and 30-40% (right). Black data points depict all $p{\pi }^{-}$ pair distributions and are fitted with the Gaussian function in addition to the combinatorial background distributions depicted in the blue lines, which were estimated via side-band fitting.

To ensure that the KF Particle method can be reliably used for extracting the physical yields, we applied the KF Particle method to a Monte Carlo simulated sample generated using an embedding technique detailed as follows: Simulated Λ particles with flat p_T and rapid distributions were propagated through a GEANT3 [25] simulation of the STAR TPC. The Λ particles decayed inside the simulated detector and the electronic signals originating from the decay particles were mixed with those from a given event from the real data. The number of simulated Λ particles was 5% of the measured charged-particle multiplicity of the event in which the simulated particles were embedded, and the simulated Λ particles all originated from the primary vertex of that event. The combined electronic signals were subsequently processed using the STAR tracking software, which is used for real data processing as well. Thereafter, the KF Particle package was deployed on the resultant tracks for Λ reconstruction.

We compared the performance of KF particles on real data and MC simulation samples. The topological variables listed below (Table 1) were used to select the Λ candidates during the KF Particle reconstruction.

Statistical criteria were used instead of geometric quantities correspondingly ($DC{A}_{\text{prim}\mathrm{,}\pi }\to {\chi }_{\text{prim}\mathrm{,}\pi }^2$, $DC{A}_{\text{prim}\mathrm{,}p}\to {\chi }_{\text{prim}\mathrm{,}p}^2$, $DC{A}_{p-\pi }\to {\chi }_{p-\pi }^2$, b and $\theta \to {\chi }_{\text{topo}\mathrm{,}\Lambda }^2$, $d_{\Lambda }\to d_{\Lambda }/\sigma d_{\Lambda }$). Comparisons of these variables between the data and MC simulation for Λ candidates with p_T=0.4-1.2 GeV/c and a centrality between 0-10% are depicted in Fig. 4. In general, the distributions of these topological variables from the data are appropriately described by the MC simulations for all centralities and p_T.

Fig. 4Full-screen View

Key topological variable distributions: (a) ${\chi }_{\text{prim}\mathrm{,}\pi }^2$, (b) ${\chi }_{\text{prim}\mathrm{,}p}^2$, (c) ${\chi }_{\text{topo}\mathrm{,}\Lambda }^2$, (d) ${\chi }_{p-\pi }^2$, (e) d_Λ, (f) $d_{\Lambda }/\sigma d_{\Lambda }$ used in KF Particle method for Λ reconstruction. Data (black points) and MC simulations (red curves) are compared.

To achieve the optimal significance of the Λ signal, the Toolkit for Multivariate data A analysis is used. TMVA is a family of supervised learning algorithms that can be used to differentiate between signals and backgrounds. For further details, please refer to Refs. [20]. Signal and background samples were prepared as inputs for training. The signal samples were obtained from a GEANT3 simulation as described above. For the background sample, we selected sidebands ($3\sigma <|m_{p\pi }-m_{\Lambda \mathrm{,}\text{PDG}}|<6\sigma$) pπ pairs in the real data around the Λ mass peak, where σ is the width of the Λ mass peak and mpπ and $m_{\Lambda \mathrm{,}\text{PDG}}$ are the masses of the pπ pair and the Λ baryon from the PDG respectively. These signal and background samples are further divided into different pT and centrality classes. We used the Boosted Decision Tree method for training. Decision-tree learning uses a set of input features and splits the input data recursively based on these features. In our case, the input features were the topological variables listed in Tab. 1 and the input data are the signal and background samples depending on these variables. BDT combines multiple decision trees to strengthen the differentiation power; a detailed discussion can be found in Ref. [26]. Training considers the correlations between different topological variables and collapses them into a single value, referred to as the BDT response value.

The BDT response value distributions from the signal and background samples for Λ candidates with p_T=0-1 GeV/c and centrality 0-10% are shown in the left panel of Fig. 5. We observe that the BDT response values for the signal and background are significantly different from each other, and thus serve as a good measure for differentiating between the signal and background. To select a BDT response cut value to optimize the significance $S/\sqrt{S+B}$, where S stands for signal counts and B stands for background counts, we used the TMVA package to calculate the signal and background efficiency as a function of the BDT response cut value, ${\epsilon }_{\text{S}}\left(\text{BDT cut}\right)$ and ${\epsilon }_{\text{B}}\left(\text{BDT cut}\right)$, using the signal and background samples, respectively. The signal and background efficiencies for Λ candidates in the p_T range 0-1 GeV/c and a centrality 0-10% are shown in the right panel of Fig. 5 for blue and red lines. The estimated significance as a function of the BDT cut-off value can then be calculated using Eq. 3: $Sig\mathrm{.}\left(\text{BDT cut}\right)=\frac{S_0{\epsilon }_{\text{S}}}{\sqrt{S_0{\epsilon }_{\text{S}}+B_0{\epsilon }_{\text{B}}}}\mathrm{,}$ (3) where S₀ and B₀ are the number of signal and background counts in the dataset before the BDT cut is applied. S₀ is obtained from the estimated Λ counts without performing any cut on the topological variables, and B₀ is obtained from the number of sideband $p{\pi }^{-}$ pairs without the BDT cut. The calculated significance as a function of the cut value applied to the BDT response value for Λ candidates in the p_T range 0-1 GeV/c and 0-10% centrality is shown in the right panel of Fig. 5 as a green line. We find that a cut value of -0.09 maximizes the significance of this particular p_T and centrality bin, and we choose this cut value for Λ reconstruction. This procedure is then repeated for each p_T and the centrality bin. Generally, as the signal-to-background ratio decreases, a stricter BDT selection cut is necessary to optimize the significance.

Fig. 5Full-screen View

(Color online) (left) BDT response value distributions for signal (blue) and background (red) Λ candidates in the p_T range 0-1 GeV/c for 0-10% centrality. (right) Efficiency for signal (blue line) and background (red line) Λ candidates in the p_T range 0-1 GeV/c for 0-10% centrality as a function of the cut value placed on the BDT response value. Significance (green line) achieves its maximum value when the cut value is -0.09.

We extracted the number of signals and background counts for each pT and the centrality bin using the tuned BDT cuts obtained, as explained above. We then used the standard helix swimming method used in previous STAR analyses [6], tuned the topological cuts in the HS method by the BDT, extracted the corresponding number of signals and background counts using the same procedure, and compared the significance obtained using these two methods. For a fair comparison, the track quality and particle identification cuts were identical. The ratios of significance as functions of pT for the three centrality selections are shown in Fig. 6. The increase in significance is approximately independent of the centrality, ≈30% in the pT range 1-3 GeV/c, and increases at low pT to ≈50%. This demonstrates that the KF Particle method is more significant for Λ signal extraction in Au+Au collisions at $\sqrt{s_{\text{NN}}}=27$ GeV in the STAR experiments.

Fig. 6Full-screen View

(Color online) Ratio of significance for Λ particles using the KF Particle method in conjunction with BDT training over those using the helix swimming (HS) method in conjunction with BDT training as a function of p_T of the Λ particles for centrality selection 0-5% (black), 30-40% (red) and 60-80% (blue). The shaded bands indicate the statistical uncertainties.

Ω Reconstruction

Next, we turn to Ω baryon. Ω baryons were reconstructed using the decay channel $\Omega \to \Lambda +K^{-}\to p+{\pi }^{-}+K^{-}$. Ω particles decayed with a proper decay length of cτ ≈ 25 mm [4], and the Λ daughters decayed again soon thereafter. The final daughter tracks were detected using STAR TPC. Similarly, for each proton, kaon or pion track, a minimum of 15 hits were required to ensure good track quality. We reconstruct the Λ baryons with the KF Particle method first and then treat it as a daughter track to reconstruct the Ω production vertex. In Fig. 7, clear Ω mass peaks were observed in the $\Lambda K^{-}$ invariant mass distributions using the KF Particle method.

Fig. 7Full-screen View

$\Lambda K^{-}$ invariant mass distributions of p_T=1.2-1.6 GeV/c in Au+Au collisions at $\sqrt{s_{\text{N}N}}=27$ GeV with centrality 0-5% (left), and 30-40% (right). Black data points depict all $\Lambda K^{-}$ pair distributions and are fitted with the Gaussian function plus the combinatorial background distributions shown in the blue lines, which are estimated via side-band fitting.

Because the decay topology for Ω baryons is more complicated than that for Λ baryons, more topological variables can be used for training to facilitate the differentiation between the signal and background. The topological variables are listed in Tab. 2 were used in the selection of Ω baryon candidates during KF Particle reconstruction.

Similar to the Λ baryon study, we generated an MC sample of the reconstructed Ω baryons using a GEANT3 simulation of the STAR TPC. The data-MC comparison of key topological variables is shown in Fig. 8.

Fig. 8Full-screen View

Key topological variable distributions: (a) ${\chi }_{\text{topo}\mathrm{,}\Lambda }^2$, (b) ${\chi }_{p-\pi }^2$, (c) ${\chi }_{\text{topo}\mathrm{,}\Omega }^2$, (d) ${\chi }_{\Lambda -K}^2$, (e) $d_{\Lambda }/{\sigma }_{d_{\Lambda }}$, (f) d_Ω/σd_Ω used in KF Particle method for Ω reconstruction. Data (black points) and MC simulations (red curves) are compared.

We find reasonable agreement between the data and MC simulations, which suggests proper estimation and usage of the covariance matrix of the Λ daughters and gives us confidence that the KF Particle method may be reliably used for the extraction of Ω baryon yields. We then generated a signal and background sample using the same method as in Λ analysis to supply inputs for TMVA training using the BDT method. The BDT response value distribution for Ω candidates with p_T=1-4 GeV is shown in the left panel of Fig. 9. The signal efficiency, background efficiency, and significance are shown in the right panel of Fig. 9. As in the case of Λ analysis, we selected the BDT response cut-off value that optimizes significance.

Fig. 9Full-screen View

(Color online) (left) BDT response value distributions for signal (blue) and background (red) Ω candidates in the p_T range 1-4 GeV/c for 0-10% centrality. (right) Efficiency for signal (blue line) and background (red line) Ω candidates in the p_T range 1-4 GeV/c for 0-10% centrality as a function of the cut value placed on the BDT response value. The significance (green line) achieves its maximum value when the cut value is 0.09.

This process is repeated for each p_T and the centrality bin. The significance of using the optimized BDT response cuts for each p_T and centrality bin was extracted. We then performed signal extraction using the default helix swimming method, with candidate selection cuts chosen to be the same as in the previous Ω analyses at the same collision energy [6, 27], also tuned by the BDT method. The signal and background counts were extracted using the default helix swimming method, and the ratios of the significances were calculated using these two methods, as shown in Fig. 10. We observe an ≈50% increase in significance in the p_T range of 1-4 GeV/c. This increase is higher than that for Λ, likely owing to the more complex decay topology with two decay vertices reconstructed by KF particles and a larger background. Further studies using KF particles are underway to extend the Ω measurement to low p_T beyond 1 GeV/c; however, this is beyond the scope of this study.

Fig. 10Full-screen View

(Color online) Ratio of significance for Ω particles using the KF Particle method in conjunction with BDT training over those using the helix swimming(HS) method in conjunction with BDT training as a function of p_T of the Ω particles for centrality selection 0-5% (black), 30-40% (red) and 60-80% (blue). The shaded bands indicate the statistical uncertainties.

D⁰ Reconstruction

D⁰ (and ${\overline{D}}^0$) particles were reconstructed via the decay channel $D^0\to K^{-}{\pi }^{+}$ and its charge conjugation with an appropriate decay length of cτ ≈ 123 μM [4]. Because this decay length is less than the spatial resolution of the TPC detector, information from the microvertex detector HFT is used to distinguish the D⁰ decay vertex from the primary collision vertex. For each kaon or pion daughter track, a minimum of 15 hits in the TPC and a match with the HFT detector with at least three hits were required to ensure good track quality. For kaon and pion particle identification, in addition to the requirement that $\left|n{\sigma }_{\pi }\right|<3$ and $\left|n{\sigma }_{\text{K}}\right|<2$, we also utilized information from the time-of-flight (TOF) detector [5] when available to help identify the particles at high p_T by requiring the measured inverse velocity (1/β) to be within three standard deviations The topological variables are listed in Table 3 were used to select the D⁰ meson candidates in KF Particle reconstruction. p_T</>,π and pT,K cuts are added to reject combinatorial background at low p_T.

Similar to the Λ and Ω baryon studies, we generated an MC sample of reconstructed D⁰ mesons using a GEANT3 simulation of the STAR TPC, HFT, and TOF and processed it through full detector tracking, as was done in the real data reconstruction with the previously mentioned embedding technique. The HFT simulator was tuned to reproduce the single-track efficiency and DCA pointing resolution observed in real data. However, the consistency in the topological variable distributions between the data and MC for D⁰ signals is yet to be demonstrated. Fig. 11 shows a comparison of several key topological variables used in the KF Particle method for D⁰ reconstruction between the data (black data points) and MC (red histograms). We found good agreement between the data and MC simulations for these variables, which means that this multiple-detector-combined MC simulation can generate D⁰ signals reasonably. The background distributions are shown in Fig. 11 (blue circles)). They are estimated from real data using the sideband method, in which background candidates are selected by requiring an invariant mass of Kπ pairs within $3\sigma <|M_{\text{i}nv}-M_{D^0}|<6\sigma$ (σ is the Gaussian width of the D⁰ signal). The signal and background behaviors can be distinguished well, particularly from $d_{D^0}/{\sigma }_{d_{D^0}}$ and ${\chi }_{\text{topo}\mathrm{,}{D}^0}^2$ distributions.

Fig. 11Full-screen View

(Color online) Key topological variables distributions: (a) ${\chi }_{\text{prim}\mathrm{,}\pi }^2$, (b) ${\chi }_{\text{prim}\mathrm{,}K}^2$, (c) ${\chi }_{K\mathrm{,}\pi }^2$, (d) $d_{D^0}/{\sigma }_{d_{D^0}}$, (e) ${\chi }_{\text{topo}\mathrm{,}{D}^0}^2$ used in KF Particle method for D⁰ reconstruction. Data (black points), MC simulations (red curves), and background (blue circles) are compared.

Thereafter, we used the signal sample generated from the MC simulation and the background sample from the sideband candidates in the data to conduct TMVA training with the BDT method to determine the topological selection working point for the best signal significance. Figure 12 left panel exhibits the BDT response value distributions for the D⁰ signal and background in the region of p_T=2-3 GeV/c with a centrality 10-40%; the efficiencies of the signal and background are also shown in the right panel. The significance was normalized to the maximum value. We determine the BDT response cut value to optimize the significance of D⁰ for each p_T and centrality class.

Fig. 12Full-screen View

(Color online) (left) BDT response value distributions for signal (blue) and background (red) D⁰ candidates in the p_T range 2-3 GeV/c for 10-40% centrality. (right) Efficiency for signal (blue line) and background (red line) D⁰ candidates in the p_T range 2-3 GeV/c for 10-40% centrality as a function of the cut value placed on the BDT response value. The significance (green line) achieves its maximum value when the cut value is 0.05.

Thereafter, we applied the optimized BDT selection cuts to real data analysis. Figure 13 displays the D⁰-invariant mass distributions derived using the KF Particle method for 10-40% Au+Au collisions at $\sqrt{s_{\text{N}N}}=200$ GeV in the regions p_T=0-1 GeV/c (left) and p_T=1.5-2.0 GeV/c (right), respectively. Black lines depict the function fits of the data with a Gaussian function for the D⁰ signal and linear background. Compared with the Λ and Ω reconstructions, more combinatorial backgrounds are included because the D⁰ decay points are closer to the collision points and blend with the particles from them, especially at p_T=0-1 GeV/c.

Fig. 13Full-screen View

$K^{\pm }{\pi }^{\mp }$ invariant mass distributions using the KF Particle method in 10-40% Au+Au collisions at $\sqrt{s_{\text{N}N}}=200$ GeV in the region of p_T=0-1 GeV/c (left) and p_T=1.5-2.0 GeV/c (right). Black data points depict all $K^{\pm }{\pi }^{\mp }$ pair distributions and are fitted with the Gaussian function plus the combinatorial background distributions shown in the blue lines, which are estimated via side-band fitting.

Thereafter, the signal significance was calculated from the invariant mass distributions of D⁰ candidates. The signal counts were obtained from a Gaussian function fit of the D⁰ peak, whereas the background counts were determined based on a linear background function fit within a mass window of $|M_{\text{i}nv}-M_{D^0}|<3\sigma$, where σ denotes the width of D⁰ peak. We compared the significance of D⁰ from the KF Particle method to the helix swimming (HS) method used in a previous analysis [5] which was in conjunction with BDT training; the ratio of the D⁰ significance between these two methods is illustrated in Fig. 14. The shaded bands indicate the statistical uncertainties in this calculation. This comparison demonstrates that the KF Particle method improves the reconstructed D⁰ signal significance, especially for low p_T and more central collisions. In 0-10% of the central Au+Au collisions at p_T<1 GeV/c, the improvement can signify as a factor of ≈3, potentially because of the enormous amount of combinatorial background (hundreds of signals) in that particular range, and the cuts based on statistical criteria work appropriately in discriminating the particles originating from a secondary vertex.

Fig. 14Full-screen View

(Color online) Ratio of significance for D⁰ particles using the KF Particle method in conjunction with BDT training over those using the helix swimming(HS) method in conjunction with BDT training as a function of p_T of the D⁰ particles for centrality selection 0-10%(black circles), 10-40%(red squares), and 40-80%(blue triangles). Shaded bands indicate the statistical uncertainties.