Charged-particle multiplicity and pseudorapidity distributions measured with the PHOBOS detector in $\text{Au}+\text{Au}$, $\text{Cu}+\text{Cu}$, $\mathbf{d}+\text{Au}$, and $\mathbf{p}+\mathbf{p}$ collisions at ultrarelativistic energies

Pseudorapidity distributions of charged particles emitted in $\text{Au}+\text{Au}$, $\text{Cu}+\text{Cu}$, $d+\text{Au}$, and $p+p$ collisions over a wide energy range have been measured using the PHOBOS detector at the BNL Relativistic Heavy-Ion Collider (RHIC). The centrality dependence of both the charged particle distributions and the multiplicity at midrapidity were measured. Pseudorapidity distributions of charged particles emitted with $\left|\eta \right|<5.4$, which account for between 95% and 99% of the total charged-particle emission associated with collision participants, are presented for different collision centralities. Both the midrapidity density ${\mathit{dN}}_{\text{ch}}/d\eta$ and the total charged-particle multiplicity $N_{\text{ch}}$ are found to factorize into a product of independent functions of collision energy, $\sqrt{s_{{}_{\mathit{NN}}}}$, and centrality given in terms of the number of nucleons participating in the collision, $N_{\text{part}}$. The total charged particle multiplicity, observed in these experiments and those at lower energies, assumes a linear dependence of $(\ln s_{{}_{\mathit{NN}}}){}^2$ over the full range of collision energy of $\sqrt{s_{{}_{\mathit{NN}}}}=2.7$–200 GeV.

Figure 1

Position and orientation of Be beam pipe and active areas of several of the detectors used in the present work. See text for details.

Figure 2

Geometrical acceptances of the Ring (light), Octagon (medium), and Vertex (dark) detectors shown as a function of pseudorapidity $\eta$ and azimuthal angle $\varphi$ for particles emitted from the nominal interaction point at the center of the Octagon array.

Figure 3

Illustration of the conversion from paddle signal to $N_{\text{part}}$ for 200 GeV $\text{Au}+\text{Au}$ collisions. Panel (a) shows the ranges of the Paddle signals corresponding to the 0%–3%, 20%–25%, and 45%–50% most central collision. The corresponding distributions of the number of participants, $N_{\text{part}}$, are shown in panel (b).

Figure 4

(a) Energy deposition spectra for $2.5<\eta <3.2$ for the Octagon detector. (b) Same as (a) but for $\left|\eta \right|<1$. The dotted histogram corresponds to the raw spectrum, whereas the solid histogram shows the effect of the signal merging correction.

Figure 5

(a) Map of the raw energy deposition versus pseudorapidity $\eta$ for the Octagon detector. (b) Same as (a) after applying the merging of signals from particle traversing multiple pads and correcting for the angle of incidence.

Figure 6

(a) Map of simulated secondary particle sources near the horizontal plane, $y=0$. For clarity, interactions in the air were neglected. The beam pipe with flanges, silicon sensors, and elements of their support structures are visible. (b) The ratio of hits from secondary particles to all hits in the multiplicity sensors located at different positions along the $z$ axis. The values are averaged over the azimuthal angle $\varphi$. The farthest Ring counters are not shown.

Figure 7

Illustration of tracklet counting using two consecutive Si layers, in this case Vertex detectors. Hits in the inner and outer Si layers are connected to form “tracklets” that point back to the vertex position in the center of the figure.

Figure 8

Illustration of the combinatorial background subtraction for non-vertex tracks for (a) Vertex detector and (b) Spectrometer tracklet analysis. The solid points represent the actual detector geometry, whereas the open points were obtained after a 180${}^{ˆ}$ rotation of one of the detector layers around the beam axis in order to estimate the combinatorial background in the acceptance region represented by vertical dash-dotted lines.

Figure 9

Single-event energy-deposition map in $\varphi$ versus $\eta$ for a central $\text{Au}+\text{Au}$ collision at 130 GeV. The open areas at $\varphi =0$ and $\varphi =\pi$ are openings in the Octagon detector in front of the Spectrometers. Small open areas at $\varphi =\pm \pi /2$ and $\eta$ about $-2$ and 1 in front of the Vertex detectors are also visible.

Figure 10

(a) The single-pad occupancy-correction factor $O(\eta )$ for central (0%–6%) and peripheral (45%–55%) $\text{Au}+\text{Au}$ collisions at 130 GeV are shown as solid and open symbols, respectively. The data for the Octagon detector are shown as circles, whereas the triangles, diamonds, and squares correspond to Ring detectors at increasing distances from the vertex position. The background-correction $f_{\text{bkg}}(\eta )$ factor and the total-correction factor $O(\eta )\times f_{\text{bkg}}(\eta )$ are shown in panels (b) and (c), respectively, using the same symbols.

Figure 11

(a) Energy deposition per detector pad $\Delta E_{\text{pad}}$, (b) energy deposition per particle track $\Delta E_{\text{trk}}$, (c) fraction of primary tracks $f_{\text{pri}}$, and (d) absorption correction factor $f_{\text{abs}}$ shown as a function of $\eta$ for central (0%–6%) (solid symbols) and peripheral $\text{Au}+\text{Au}$ collisions (45%–55%) (open symbols) at 130 GeV. The circles and squares refer to simulations for the Octagon and Ring detectors, respectively.

Figure 12

The energy dependence of the participant-scaled charged-particle multiplicity ${\mathit{dN}}_{\text{ch}}/d\eta /\langle N_{\text{part}}/2\rangle$ at midrapidity $\left|\eta \right|<1$ is shown for 0%–6% central nucleus-nucleus collisions (solid symbols) and compared to $\mathrm{p̄}p/\mathit{pp}$ non-single diffractive (NSD) (open diamonds) [42, 43, 44, 45], inelastic (open squares) [43, 44, 45, 46, 47, 48], and present work (crosses). The solid squares and triangles represent results from the PHOBOS experiment for $\text{Au}+\text{Au}$ and $\text{Cu}+\text{Cu}$ collisions, respectively. The solid circles and solid diamonds are obtained from SPS $\text{Pb}+\text{Pb}$ [22] and AGS $\text{Au}+\text{Au}$ [49] collisions, respectively. The solid triangle (pointing down) is from LHC/ALICE $\text{Pb}+\text{Pb}$ collisions [50]. The 56 GeV $\text{Au}+\text{Au}$ data point is from Ref. [2]. The solid line is a linear fit [as a function of $\sqrt{s_{\mathit{NN}}}$, Eq. (7)] to the $\text{Au}+\text{Au}$ and $\text{Pb}+\text{Pb}$ data up to 200 GeV, whereas the dash-dotted curve represents the form suggested in Ref. [50]. The dashed curve is a fit to the inelastic $\mathrm{p̄}p/\mathit{pp}$ data. See text for details.

Figure 13

Panel (a): Charged-particle multiplicity at midrapidity $\left|\eta \right|<1$ per participant pair, ${\mathit{dN}}_{\text{ch}}/d\eta /\langle N_{\text{part}}/2\rangle$, obtained from the “tracklet” analysis shown as a function of $\langle N_{\text{part}}\rangle$ (solid symbols) for $\text{Au}+\text{Au}$ collisions. The shaded ovals indicate the 90%-confidence-limit systematic errors. For comparison, open points show the “single-Si-layer” analysis. The solid curves represent fits to the data using the form given in Eqs. (10, 11, 12) (excluding the $\mathit{pp}$ points). The $\mathit{pp}$ points at $\langle N_{\text{part}}\rangle =2$ were interpolated using the fit to the inelastic $\mathit{pp}/\stackrel{̲}{p}p$ data; see Fig. 12. See text for a discussion of panel (b).

Figure 14

Same as Fig. 13, but for $\text{Cu}+\text{Cu}$ collisions.

Figure 15

${\mathit{dN}}_{\text{ch}}/d\eta$ vs $\eta$ (solid points) for ten centrality bins representing 45% of the total cross section for $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{{}_{\mathit{NN}}}}=19.6$ GeV. The solid curve is a fit to the data within the $-3<\eta <3$ region using Eq. (14) (see text for details). The shaded band represents 90%-confidence-limit systematic errors. The open points were obtained by the tracklet analysis in the range $\left|\eta \right|<1$.

Figure 16

${\mathit{dN}}_{\text{ch}}/d\eta$ vs $\eta$ (solid points) for eleven centrality bins representing 50% of the total cross section for $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{{}_{\mathit{NN}}}}=62.4$ GeV. The solid curve is a fit to the data within the $-4.2<\eta <4.2$ region using Eq. (14) (see text for details). The shaded band represent 90%-confidence-limit systematic errors. The open points were obtained by the tracklet analysis in the range $\left|\eta \right|<1$.

Figure 17

Same as Fig. 16 but for $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{{}_{\mathit{NN}}}}=130$ GeV. The solid curves represent best fits to the data over the region $-4.9<\eta <4.9$ using Eq. (14) (see text) and the shaded regions represent the systematic error band at 90% confidence limit. The open points were obtained by the tracklet analysis in the range $\left|\eta \right|<1$.

Figure 18

Same as Fig. 17 but for $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{{}_{\mathit{NN}}}}=200$ GeV. The solid curves represent best fits to the data over the full $\eta$ range using Eq. (14) and the shaded regions represent 90%-confidence-limit systematic errors. The open points were obtained by the tracklet analysis in the range $\left|\eta \right|<1$.

Figure 19

${\mathit{dN}}_{\text{ch}}/d\eta$ versus $\eta$ (solid points) for twelve centrality bins representing 55% of the total cross section for $\text{Cu}+\text{Cu}$ collisions at $\sqrt{s_{{}_{\mathit{NN}}}}=22.4$ GeV. The solid curve is a symmetric double quasi-Gaussian function fit to the data within the $-3.2<\eta <3.2$ region. The shaded band represent 90%-confidence-limit systematic errors. The open points were obtained by the tracklet analysis in the range $\left|\eta \right|<1$.

Figure 20

${\mathit{dN}}_{\text{ch}}/d\eta$ versus $\eta$ (solid points) for twelve centrality bins representing 55% of the total cross section for $\text{Cu}+\text{Cu}$ collisions at $\sqrt{s_{{}_{\mathit{NN}}}}=62.4$ GeV. The solid curve is a fit to the data within the $-4.2<\eta <4.2$ region using Eq. (14). The shaded band represent 90%-confidence-limit systematic errors. The open points were obtained by the tracklet analysis in the range $\left|\eta \right|<1$.

Figure 21

Same as Fig. 20 but for $\sqrt{s_{{}_{\mathit{NN}}}}=200$ GeV. The solid curve is a fit to the data over the full $\eta$ region using Eq. (14). The open points were obtained by the tracklet analysis for events in the range $\left|\eta \right|<1$.

Figure 22

Illustration of the methods used for estimating the total number of charged particles from the measured ${\mathit{dN}}_{\text{ch}}/d\eta$ versus $\eta$ (solid points) distributions. The solid curve represents a fit to the data within the region $-y_{\text{beam}}<\eta <y_{\text{beam}}$ using the functional form of Eq. (14). The open circles and the shaded regions are explained in the text.

Figure 23

Illustration of the trapezoidal approximation for 200-GeV $\text{Au}+\text{Au}$ collisions at 0%–3% centrality. Open circles represent the data (errors are not shown for clarity) and the black lines represent the piecewise linear trapezoidal approximation to the data as described in the text.

Figure 24

Total charged-particle multiplicity $N_{\text{ch}}$, total charged-particle multiplicity per participant pair $2N_{\text{ch}}/\langle N_{\text{part}}\rangle$, and full width at half maximum (FWHM) are shown as a function of centrality for each energy for $\text{Au}+\text{Au}$ collisions. The first two quantities are shown for both the measured region $N_{\text{ch}}|{}_{\left|\eta \right|<5.4}$(open points) and based on extrapolations, $N_{\text{ch}}^p$ (solid circles), see text for details. Corresponding data for inelastic $\mathrm{p̄}p$ collisions [43] at 200 GeV and $\mathit{pp}$ collisions [44] are shown as solid squares, whereas the crosses are from the present work and solid diamonds represent an overall fit $N_{\text{ch}}=-0.42+4.69s^{0.155}$ for inelastic collisions [52]. The FWHM for 200-GeV $\mathrm{p̄}p$ and 62.4-GeV $\mathit{pp}$ collisions (solid squares) were obtained from Refs. [43] and [44], respectively, whereas the value (solid diamond) for 19.6 GeV represents an extrapolation based on data in the latter work. No FWHM data are available for 130 GeV $\mathit{pp}$ collisions. Error bars and ellipses represent systematic 90%-confidence limit errors.

Figure 25

Same as Fig. 24 but for $\text{Cu}+\text{Cu}$ collisions. The data for 22.4-GeV $\mathit{pp}$ collisions (solid diamonds) were obtained as for the 19.6-GeV data shown in Fig. 24.

Figure 26

Doubly logarithmic (a) and linear (b) plots of $N_{\text{ch}}^p/\langle N_{\text{part}}/2\rangle$ versus $(\ln s){}^2$. The PHOBOS results are for 0%–3% central collisions. The trend is extended to lower energies with data from the SPS [22] and AGS [49]. The dashed and solid curves were calculated using Eqs. (16) and (17), respectively.

Figure 27

Two estimates of the total charged-particle multiplicity per participant pair, $2N_{\text{ch}}|{}_{\left|\eta \right|<5.4}/\langle N_{\text{part}}\rangle$ (open points) and $2N_{\text{ch}}^p/\langle N_{\text{part}}\rangle$ (solid points), are shown as a function of centrality for each energy for $\text{Au}+\text{Au}$ (a) and $\text{Cu}+\text{Cu}$ (c) collisions. Horizontal lines represent centrality-independent levels predicted by Eq. (17). The lower panels show the ratio of data to the Eq. (17) prediction. The solid line is the average of all data points and the gray-shaded area the 1$\sigma$ standard deviation. Corresponding data for $p\mathrm{p̄}/\mathit{pp}$ collisions are shown as stars, see Figs. 24 and 25 for details.

Figure 28

Charged-particle multiplicity ${\mathit{dN}}_{\text{ch}}/d\eta$ shown for five centrality bins for $d+\text{Au}$ collisions [panels (a)–(e)]. The gray-shaded area represents the 1$\sigma$ standard deviation. The minimum-bias distribution is shown in panel (f). The pseudorapidity $\eta$ is given relative to the deuteron beam, which travels in the positive $z$ direction.

Figure 29

Charged-particle multiplicity ${\mathit{dN}}_{\text{ch}}/d\eta$ shown for 200-GeV (a) and 410-GeV (b) $\mathit{pp}$ inelastic collisions. The gray-shaded area represents the 90%-confidence-limit systematic error. The solid curves are symmetric double-Gaussian fits to the data used to derive the total multiplicities listed in Table .

Figure 30

Illustration of the extended longitudinal scaling (also known as limiting fragmentation scaling) for $\text{Au}+\text{Au}$ [panels (a) and (b)], $\text{Cu}+\text{Cu}$ [panels (c) and (d)], and $p+p$ collisions [panels (e) and(f)]. The systematic errors (90% confidence limit) are shown as shaded areas for the highest energy only for each system. The arrows indicate the location of midrapidity ($\eta =$ 0).