Proton-number fluctuations in $\sqrt{{s}_{NN}}=2.4 \mathrm{GeV} \mathrm{Au} + \mathrm{Au}$ collisions studied with the High-Acceptance DiElectron Spectrometer (HADES)

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Proton-number fluctuations in $\sqrt{s_{N N}} = 2.4 GeV Au + Au$ collisions studied with the High-Acceptance DiElectron Spectrometer (HADES)

J. Adamczewski-Musch et al. (HADES Collaboration)

Phys. Rev. C 102, 024914 – Published 24 August 2020

Abstract

We present an analysis of proton-number fluctuations in $\sqrt{s_{N N}} = 2.4 GeV$ ${}^{197}{Au} + {}^{197}{Au}$ collisions measured with the High-Acceptance DiElectron Spectrometer (HADES) at GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt. With the help of extensive detector simulations done with Isospin Quantum Molecular Dynamics (IQMD) transport model events including nuclear clusters, various nuisance effects influencing the observed proton cumulants have been investigated. Acceptance and efficiency corrections have been applied as a function of fine-grained rapidity and transverse momentum bins, as well as considering local track density dependencies. Next, the effects of volume changes within particular centrality selections have been considered and beyond-leading-order corrections have been applied to the data. The efficiency and volume-corrected proton number moments and cumulants $K_{n}$ of orders $n = 1, 2, 3, and 4$ have been obtained as a function of centrality and phase-space bin, as well as the corresponding correlators $C_{n}$ . We find that the observed correlators show a power-law scaling with the mean number of protons, i.e., $C_{n} \propto {〈 N 〉}^{n}$ , indicative of mostly long-range multiparticle correlations in momentum space. We also present a comparison of our results with Au $+$ Au collision data obtained at the Relativistic Heavy Ion Collider (RHIC) at similar centralities but higher $\sqrt{s_{N N}}$ .

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Received 21 February 2020
Accepted 9 June 2020

DOI:https://doi.org/10.1103/PhysRevC.102.024914

Physics Subject Headings (PhySH)

Particle correlations & fluctuations Relativistic heavy-ion collisions

Nuclear Physics

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Vol. 102, Iss. 2 — August 2020

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Figure 1
Schematic explosion view of the HADES experiment. The forward wall hodoscope (FWALL) used for centrality selection (see text) is not shown on this picture.
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Figure 2
Distribution of the total number of hits in the HADES time-of-flight detectors TOF and RPC in the 43% most central $Au + Au$ events accepted by the PT3 trigger. The tiny contribution of order $\leq 2 \times 10^{- 4}$ , visible for $N_{hit} > 260$ , is attributed to event pileup.
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Figure 3
Total number of charged hits in the time-of-flight detectors (TOF and RPC) as a function of the reconstructed event vertex $v_{z}$ along the beam axis $z$ . The 15 gold target segments are clearly separated from the diamond START detector as well as from one another; the horizontal solid line indicates the vertex cut used to select reactions in the targets. Note that the faint high- $N_{hit}$ tail ( $N_{hit} > 100$ ) visible for the START detector is due to $Au + Au$ reactions in its thin gold platings.
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Figure 4
Ratio of transverse to longitudinal energy $E_{R A T}$ as a function of the reconstructed event vertex along the beam axis $z$ . The horizontal dashed line indicates the $E_{R A T} > 0.55$ cut used to further suppress Au $+$ C reactions in the Kapton strips of the target holder. The solid line delimits the vertex cut applied on the target region.
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Figure 5
Sum signal $Σ Q_{FW}$ measured in the forward wall vs number of hits $N_{hit}$ in the TOF and RPC. The black long-dashed line indicates a loose 2D cleaning cut applied to remove the contamination from peripheral events. Centrality selections, e.g., in steps of 5%, were done by applying additional, more restrictive cuts on $Σ Q_{FW}$ as indicated by the set of short-dashed horizontal lines.
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Figure 6
Particle identification (PID) based on particle velocity vs momentum correlations as well as on energy loss vs momentum correlations. Shown in panel (a) is $β = v / c$ of the detected particle, with velocity $v$ obtained from the time of flight, as a function of its $p / q$ , where $p$ is the particle momentum and $q$ is its charge. Note that the weak branch lying between $p$ and $d / α$ are ${}^{3}{He}$ nuclei. The dashed curves delineate the condition used to select protons. The weak ${}^{3}{He}$ contamination remaining in this selection is suppressed by applying an additional condition on the energy-loss relation, $d E / d x$ vs $p / q$ , shown in panel (b) for the masses fulfilling $0.6 < M / q < 1.6$ $GeV / c^{2}$ .
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Figure 7
Particle mass distribution of reconstructed tracks—as deduced from particle velocity, momentum, and energy loss—for positive particles within the phase-space bin of interest ( $y = y_{0} \pm 0.5$ and $0.4 < p_{t} < 1.6$ $GeV / c$ ). The masses selected by the proton cut (see Fig. 6) as well as the $Z = 1$ condition are shown in red. Contaminations from lower masses ( $π^{+}$ , $K^{+}$ ) and higher masses ( $d$ , $t$ , He) are below 0.1%, resulting in an overall proton purity $> 0.999$ .
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Figure 8
Identified proton yield as a function of laboratory rapidity $y$ and transverse momentum $p_{t}$ . The acceptance is constrained by the geometry of HADES ( $θ = 15 – 85^{o}$ ) and by momentum cuts ( $0.3 < p < 3$ $GeV / c$ ). Thin dashed curves indicate the polar angle, in steps of $10^{o}$ , and proton momentum, in steps of 0.5 $GeV / c$ . The dashed rectangle corresponds to the phase space selected for the fluctuation analysis: $y = y_{0} \pm 0.5$ and $0.4 < p_{t} < 1.6$ $GeV / c$ , where midrapidity $y_{0} = 0.74$ .
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Figure 9
Raw multiplicity distributions of fully reconstructed and identified protons in Au $+$ Au collisions within a given phase-space bin ( $y \in y_{0} \pm Δ y$ and $p_{t} = 0.4 - 1.6$ $GeV / c$ ) and for different centrality selections based on the observable $Σ Q_{FW}$ , using 10%-wide bins (top) or 5%-wide bins (bottom).
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Figure 10
Simulated proton detection efficiency $ε_{det}$ in one sector of HADES as a function of rapidity $y$ and transverse momentum $p_{t}$ , and for two impact parameter ranges (left, $b = 8 - 10$ fm and right, $b = 0 - 2$ fm).
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Figure 11
Simulated proton detection efficiency $ε_{det}$ as a function of the number of reconstructed tracks per sector $N_{trk}^{s}$ , shown here for three out of the 40 narrow phase-space bins used in our analysis. Efficiency points correspond to 2-fm-wide impact-parameter slices centered at $b =$ 9, 7, 5, 3, and 1 fm, respectively; curves are the polynomials adjusted to model the occupancy dependence of the efficiency.
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Figure 12
Proton response matrices of the HADES detector simulated with IQMD clustering for two centrality selections, semiperipheral (30–40%) and central (0–10%). These matrices encode the distribution of the number $N_{\det}$ of reconstructed proton tracks—within the phase-space bin of interest (here $y \in y_{0} \pm 0.5$ )—for any given number $N_{i n}$ of protons emitted into this bin.
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Figure 13
Moment expansion technique applied to proton distributions from IQMD events (0–10% most central, $y \in y_{0} \pm 0.2$ ). Shown is the dependence of the corrected moments on the truncation order $n_{\max}$ of the moment matrix $A$ . The plateau region ( $n_{\max} \geq 10$ ), corresponding to stable behavior, is close to the true IQMD values, namely $〈 N_{p} 〉 = 22.5$ (a), $σ = 6.15$ (b), skewness $\equiv γ_{1} = 0.14$ (c), and kurtosis $\equiv γ_{2} = - 0.16$ (d), indicated by red dashed lines.
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Figure 14
Comparison of various efficiency correction methods applied to a simulation using IQMD events and selecting protons within $y \in y_{0} \pm 0.2$ . Plotted are the corrected proton number mean (a), width (b), skewness (c), and kurtosis (d) as a function of $〈 N_{part} 〉$ . Error bars are statistical and colored shaded bands correspond to systematic errors of the correction technique: blue for the binomial event-by-event correction, and ocher for unfolding, respectively. The IQMD truth is shown as black dashed lines.
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Figure 15
Same as Fig. 14 but for protons within a rapidity bin of $y \in y_{0} \pm 0.5$ . Note that only the most promising techniques are shown here (see text).
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Figure 16
Reduced proton cumulants $κ_{n}$ as a function of $N_{part}$ in four different transport calculations done with the IQMD, UrQMD, and HSD models respectively. The phase space chosen here was $y \in y_{0} \pm 0.2$ (or $y \in y_{0} \pm 0.5$ ) and $0.4 \leq p_{t} \leq 1.6$ $GeV / c$ for IQMD, respectively $y \in y_{0} \pm 0.2$ and $0.4 \leq p_{t} \leq 1.6$ $GeV / c$ for UrQMD and HSD. Dashed lines are linear fits of function $κ (N_{part}) = κ_{0} + κ^{'} N_{part}$ to some of the presented reduced cumulants.
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Figure 17
Illustration of the magnitude of NLO $+$ N2LO volume-fluctuation terms contributing to the observed reduced proton number cumulants ${\tilde{κ}}_{n}$ with $n = 1, 2, 3$ , and 4. The magnitude of the terms decreases quickly with increasing order of $v_{i}$ ; low-order diagonal terms are dominant and off-diagonal terms, depending on a product $v_{i} \cdot v_{j}$ , fall off even faster. The corrections shown were determined for a selection of the 0–10% most central Au $+$ Au collisions measured in HADES (see Sec. 5 below).
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Figure 18
Correlation between the total number of hits $N_{hit}$ in the HADES time-of-flight detectors and the number of participants $N_{part}$ in $1.23 A$ GeV Au $+$ Au events simulated with the IQMD transport code for impact parameters in the range of 0–10 fm.
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Figure 19
Transport simulations of $Au + Au$ events with the IQMD (a) and UrQMD (b) models showing correlations between the number of identified protons $N_{prot}$ and the centrality defining observable $N_{cen} = N_{hit}$ , $N_{trk}$ , and $Σ Q_{FW}$ , respectively. Pearson's linear correlation coefficient $ρ (N_{prot}, N_{cen})$ is displayed for two rapidity ranges, $y \in y_{o} \pm 0.2$ (dark blue) and $y \in y_{o} \pm 0.5$ (red), as a function of the impact parameter $b$ .
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Figure 20
$N_{part}$ distributions of simulated IQMD events shown for four centrality selections based on the $Σ Q_{FW}$ signal in the FWALL. IQMD truth (red histograms) is compared to reconstructions based on their first four volume cumulants, displayed as four-parameter $β$ functions, without correlation effects (dot-dashed curves) and with correlations (solid curves). The sum curves correspond to the 0–36% most central events.
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Figure 21
(a) Measured $Au + Au$ $N_{hit}$ distributions of eight 5% centrality selections based on the $Σ Q_{FW}$ signal. The latter is constrained by the 2D cut shown in Fig. 5, causing also the barely visible “knee” in the 15–20% distribution. (b) Corresponding reconstructed $N_{part}$ distributions—plotted as four-parameter $β$ functions—when using the rescaled and modified volume cumulants $v_{l}^{mod}$ (solid curve) or the just rescaled ones $v_{l}$ (dot-dashed curve). The sum curves correspond to the 40% most central events.
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Figure 22
Au $+$ Au data: Efficiency and N2LO volume-corrected proton cumulant ratios plotted as a function of the width of the rapidity bin defined by $y \in y_{0} \pm Δ y$ and $0.4 \leq p_{t} \leq 1.6$ $GeV / c$ . Shown are $ω = K_{2} / K_{1}$ (top), $γ_{1} \times σ = K_{3} / K_{2}$ (middle), and $γ_{2} \times σ^{2} = K_{4} / K_{2}$ (bottom) for various 5% centrality selections. Error bars are statistical only, and dashed lines connect the data points belonging to a given centrality. With decreasing $Δ y$ , all ratios tend toward unity (indicated also by a horizontal line); i.e., they approach the Poisson limit where $K_{1} = K_{2} = K_{3} = K_{4}$ .
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Figure 23
Au $+$ Au data: Efficiency and volume-corrected reduced proton cumulants $κ_{n}$ for the phase-space bin $y \in y_{0} \pm 0.2$ and $0.4 \leq p_{t} \leq 1.6$ $GeV / c$ as a function of mean $N_{part}$ , using 5% centrality bins. Shown are the data without volume correction (open triangles), with Skokov et al. [77] correction (open circles), and with N2LO correction (solid squares). Vertical bars are statistical errors, cups delimit full systematic errors (shown on the N2LO corrected points only), and horizontal bars shown in panel (a) correspond to the width ( $\pm 1$ s.d.) of the $N_{part}$ distribution in the given centrality bin. Solid curves are N2LO fits, dashed curves are NLO fits (for comparison), and shaded bands are the $\pm 1$ s.d. statistical errors of the fits (orange for NLO, olive for N2LO).
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Figure 24
Au $+$ Au data: Efficiency and N2LO volume-corrected reduced proton cumulants as a function of mean $N_{part}$ , using 5% centrality bins. Data from two phase-space selections are shown: $y \in y_{0} \pm 0.2$ (blue squares) and $y \in y_{0} \pm 0.4$ (red inverted triangles), both with $0.4 \leq p_{t} \leq 1.6$ $GeV / c$ . Vertical bars are statistical errors on data, cups delimit full systematic errors; horizontal bars, shown in panel (a) only, correspond to the width ( $\pm 1$ s.d.) of the $N_{part}$ distribution in the given centrality bin. Solid curves are N2LO fits to the data and shaded bands are the $\pm 1$ s.d. statistical fit errors.
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Figure 25
Au $+$ Au data: Efficiency and N2LO volume-corrected proton cumulant ratios as a function of mean $N_{part}$ , using 5% centrality bins. Two selections are shown: $y \in y_{0} \pm 0.2$ (blue squares) and $y \in y_{0} \pm 0.4$ (red inverted triangles), both with $0.4 \leq p_{t} \leq 1.6$ $GeV / c$ . Vertical error bars are statistical, cups demarcate full systematic errors; horizontal bars, shown in panel (b) only, depict the width ( $\pm 1$ s.d.) of the $N_{part}$ distribution within the given centrality bin. Solid curves are N2LO fits, shaded bands represent the corresponding $\pm 1$ s.d. statistical fit errors. The horizontal dashed lines correspond to the Poisson limit where $K_{1} = K_{2} = K_{3} = K_{4}$ .
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Figure 26
Au $+$ Au data: Efficiency and N2LO volume-corrected proton cumulants $K_{n}$ [(a)–(c)] and correlators $C_{n}$ [(d)–(f)], shown as a function of the rapidity bin $\pm Δ y$ , i.e., a phase space $y \in y_{0} \pm Δ y$ and $0.4 \leq p_{t} \leq 1.6$ $GeV / c$ . Dashed lines connect data points for guidance only, bars are statistical errors, and cups indicate full systematic errors. To not clutter the graphs, the latter are shown only for a few of the centrality selections, i.e., 0–5%, 5–10%, and 10–15%.
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Figure 27
Au $+$ Au data: Efficiency and N2LO volume corrected proton number correlators $C_{n}$ in the rapidity bin $y \in y_{0} \pm 0.4$ as a function of the upper cut on transverse momentum $p_{t}^{max}$ , shown here for 10% centrality bins. Error bars are statistical only. Saturation of the correlators is reached for $p_{t}^{max} ≅ 1.5$ $GeV / c$ .
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Figure 28
Au $+$ Au data: Efficiency and N2LO volume-corrected proton correlators $C_{n}$ as a function of the mean number of protons $〈 N_{p} 〉$ within the selected phase-space bin, $y \in y_{0} \pm Δ y$ ( $Δ y = 0.1, ..., 0.5$ ) and $0.4 \leq p_{t} \leq 1.6$ $GeV / c$ , and for eight centrality selections. Error bars on data are statistical, cups delimit systematic uncertainties (shown, for clarity, only on the 0–5% selection). Black dashed lines connect the data points in a given centrality selection and red solid curves are power-law fits $C_{n} \propto {〈 N_{p} 〉}^{α}$ . Only a few of the fit curves are actually presented; however, all adjusted values of parameter $α$ are listed in Table 3.
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Figure 29
$Au + Au$ data: Evolution of the scaled cumulants $K_{n} / K_{2}$ as a function of center-of-mass energy $\sqrt{s_{N N}}$ for two centrality bins (0–10% or 0–5%, red symbols, and 30–40%, black symbols) and shown as $γ_{1} \times σ$ (left column) and $γ_{2} \times σ^{2}$ (right column). STAR data [27, 33] is shown for a $y_{0} \pm 0.5$ phase-space bite, and HADES data for $y_{0} \pm 0.2$ (top row) and $y_{0} \pm 0.4$ (bottom row), respectively. Vertical bars are statistical errors; full systematic errors are shown for the HADES data as shaded bars and for the STAR data points as cups.
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Figure 30
Comparison of the normalized distribution of detected particles $N_{det}$ resulting from different efficiency models: binomial, hypergeometric, $β$ -binomial, and our occupancy model (see text). In all cases, the input number of particles is $N_{in} = 40$ and the parameters of the models are adjusted to realize an average efficiency of $〈 ε 〉 = 0.6$ , resulting in $〈 N_{det} 〉 = 24$ . Calculations done with the occupancy model are shown for two segmentations: $N = 100$ , with a single-hit efficiency of $ε = 0.68$ , and $N = 50$ , with $ε = 0.81$ .
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Figure 31
Nonbinomial efficiencies of the occupancy model as a function of the particle multiplicity $m$ for $ε \equiv N Ω = 0.59$ and different values of $N$ .
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Figure 32
Illustration of the four-parameter $β$ distribution [Eq. (C3), not normalized]. The effect of nonzero skewness (a) and nonzero kurtosis (b) on the distribution shape is shown. Note that the limits $γ_{1} ≃ 0$ and $γ_{2} ≃ 0$ approximate the $N (200, 50)$ normal distribution also plotted in panel (a).
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Figure 33
Comparison of the four-parameter $β$ distribution (red line) with the true $N_{part}$ distributions (dark blue histogram) obtained from IQMD for four different centrality selections. For each selection, the two distributions are normalized to the same number of counts.
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Physical Review C

covering nuclear physics

Proton-number fluctuations in $\sqrt{s_{N N}} = 2.4 GeV Au + Au$ collisions studied with the High-Acceptance DiElectron Spectrometer (HADES)

J. Adamczewski-Musch et al. (HADES Collaboration)

Phys. Rev. C 102, 024914 – Published 24 August 2020

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Physical Review C

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Proton-number fluctuations in sNN=2.4 GeV Au + Au collisions studied with the High-Acceptance DiElectron Spectrometer (HADES)

J. Adamczewski-Musch et al. (HADES Collaboration)

Phys. Rev. C 102, 024914 – Published 24 August 2020

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Proton-number fluctuations in $\sqrt{s_{N N}} = 2.4 GeV Au + Au$ collisions studied with the High-Acceptance DiElectron Spectrometer (HADES)