System-size and centrality dependence of charged kaon and pion production in nucleus-nucleus collisions at 40$A$ GeV and 158$A$ GeV beam energy

Measurements of charged pion and kaon production are presented in centrality selected $\mathrm{Pb}+\mathrm{Pb}$ collisions at 40$A$ GeV and 158$A$ GeV beam energy as well as in semicentral $\mathrm{C}+\mathrm{C}$ and $\mathrm{Si}+\mathrm{Si}$ interactions at 40$A$ GeV. Transverse mass spectra, rapidity spectra, and total yields are determined as a function of centrality. The system-size and centrality dependence of relative strangeness production in nucleus-nucleus collisions at 40$A$ GeV and 158$A$ GeV beam energy are derived from the data presented here and from published data for $\mathrm{C}+\mathrm{C}$ and $\mathrm{Si}+\mathrm{Si}$ collisions at 158$A$ GeV beam energy. At both energies a steep increase with centrality is observed for small systems followed by a weak rise or even saturation for higher centralities. This behavior is compared to calculations using transport models (ultra-relativistic quantum molecular dynamics and hadron-string dynamics), a percolation model, and the core-corona approach.

Figure 1

Transverse mass spectra at mid-rapidity from the combined $dE/dx$ and TOF analysis for K${}^{+}$ and K${}^{-}$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at 40$A$ GeV ($-0.1<y<0.1$) and 158$A$ GeV ($-0.2<y<0.2$) beam energy. Lines show exponential fits for 0.0 GeV/$c^2$ $<m_{\text{T}}-m_0<$ 0.8 GeV/$c^2$ (40$A$ GeV) and 0.0 GeV/$c^2$ $<m_{\text{T}}-m_0<$ 1.0 GeV/$c^2$ (158$A$ GeV). Different centralities are scaled for clarity. The error bars indicate the statistical uncertainty.

Figure 2

Transverse mass spectra at mid-rapidity ($0<y<0.2$) for ${\pi }^{-}$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at 40$A$ GeV and 158$A$ GeV beam energy. Lines show exponential fits for 0.2 $<m_{\text{T}}-m_0<$ 0.7 GeV/$c^2$. Different centralities are scaled for clarity. The error bars indicate the statistical uncertainty.

Figure 3

Transverse mass spectra of ${\pi }^{-}$ ($0<y<0.2$) and kaons ($0.8<y<1.0$) at 40$A$ GeV in $\mathrm{C}+\mathrm{C}$ and $\mathrm{Si}+\mathrm{Si}$ collisions. Lines show exponential fits for 0.2 GeV/$c^2<m_{\text{T}}-m_0<$ 0.8 GeV/$c^2$ (${\pi }^{-}$) and 0.1 GeV/$c^2<m_{\text{T}}-m_0<0.5$ GeV/$c^2$ (kaons). The inverse slope parameters for ${\text{K}}^{+}$(${\text{K}}^{-}$) are $171\pm 18$ MeV for $\mathrm{C}+\mathrm{C}$ and $203\pm 10$ MeV for $\mathrm{Si}+\mathrm{Si}$ collisions ($137\pm 24$ MeV for $\mathrm{C}+\mathrm{C}$ and $166\pm 13$ MeV for $\mathrm{Si}+\mathrm{Si}$ collisions). The inverse slopes of pions are given in Table 4. The error bars indicate the statistical uncertainty.

Figure 4

Mean transverse mass as extracted from the transverse mass spectra at mid-rapidity for ${\pi }^{-}$, K${}^{+}$, and K${}^{-}$ vs number of participating nucleons. Open symbols show results from $p+p$, $\mathrm{C}+\mathrm{C}$, $\mathrm{Si}+\mathrm{Si},$ and central $\mathrm{Pb}+\mathrm{Pb}$ collisions. Except for the new results for $\mathrm{C}+\mathrm{C}$ and $\mathrm{Si}+\mathrm{Si}$ collisions at 40$A$ GeV beam energy the data are taken from [2, 3, 8, 9, 10]. Filled symbols represent the centrality selected $\mathrm{Pb}+\mathrm{Pb}$ data. The thin vertical bars indicate the statistical errors, which are mostly not visible. The thick shaded bars indicate the systematic errors.

Figure 5

Rapidity spectra for ${\pi }^{-}$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at 40$A$ GeV and 158$A$ GeV beam energy for different centrality bins labeled C0–C4. Open symbols are reflected at mid-rapidity. Solid lines represent the fits of Eq. (2) (often not visible due to symbol size). The dotted and dashed lines represent HSD [36] and UrQMD2.3 [37] simulations, respectively. The error bars (mostly not visible) indicate the statistical uncertainty.

Figure 6

Rapidity spectra of K${}^{+}$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at 40$A$ GeV and 158$A$ GeV beam energy for different centrality bins labeled C0–C4. Circles are values from the $dE/dx$ analysis; the stars give the results from the combined $dE/dx$ and TOF analyses. Open symbols are reflected at mid-rapidity. Solid lines represent the fits of Eq. (2) (often not visible due to symbol size). The dotted and dashed lines represent HSD [36] and UrQMD2.3 [37] simulations, respectively. The error bars (mostly not visible) indicate the statistical uncertainty.

Figure 7

Rapidity spectra of K${}^{-}$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions at 40$A$ GeV and 158$A$ GeV beam energy for different centrality selections labeled C0–C4. Circles are values from the $dE/dx$ analysis; the stars give the results from the combined $dE/dx$ and TOF analysis. Open symbols are reflected at mid-rapidity. Solid lines represent the fits of Eq. (2). The dotted and dashed lines represent HSD [36] and UrQMD2.3 [37] simulations, respectively. The error bars (mostly not visible) indicate the statistical uncertainty.

Figure 8

Rapidity spectra for ${\pi }^{-}$ and K${}^{\pm }$ for semicentral $\mathrm{C}+\mathrm{C}$ and $\mathrm{Si}+\mathrm{Si}$ collisions at 40$A$ GeV beam energy. Solid lines show free fits of double Gaussians. Open symbols are reflected at mid-rapidity. For K${}^{\pm }$ also double Gaussians with width $\sigma$ and shift $y_0$ as for central $\mathrm{Pb}+\mathrm{Pb}$ collisions at 40$A$ GeV beam energy were fitted (short dashed line) as well as a single Gaussian with the same $RM{S}_y$ (dashed line). The error bars indicate the statistical uncertainty.

Figure 9

Width of the rapidity distribution described by the rms values for ${\pi }^{-}$, K${}^{+}$, and K${}^{-}$ vs mean number of participating nucleons. Open symbols show results from $p+p$, $\mathrm{C}+\mathrm{C}$, $\mathrm{Si}+\mathrm{Si},$ and central $\mathrm{Pb}+\mathrm{Pb}$ collisions. Except for the new results for $\mathrm{C}+\mathrm{C}$ and $\mathrm{Si}+\mathrm{Si}$ collisions at 40$A$ GeV beam energy the data are taken from [2, 3, 8, 9]. Filled symbols represent the centrality selected $\mathrm{Pb}+\mathrm{Pb}$ data. Lines show results from HSD [36] and UrQMD2.3 [37] simulations as indicated in the figure. The thin vertical bars indicate the statistical errors, which are mostly not visible. The thick shaded bars indicate the systematic errors.

Figure 10

Particle yields divided by the mean number of wounded nucleons as a function of centrality. Closed symbols show results from centrality selected $\mathrm{Pb}+\mathrm{Pb}$ collisions; open circles show $p+p$ data from NA49 [9, 10] as well as $\mathrm{C}+\mathrm{C}$, $\mathrm{Si}+\mathrm{Si}$ (at 158$A$ GeV [8]), and central $\mathrm{Pb}+\mathrm{Pb}$ collisions from [2, 3]. The stars denote the values obtained for $p+p$ interactions from parametrization “B” in Ref. [38]. At 40 GeV$/c$ the yields of K${}^{\pm }$ in $p+p$ are compared to values extracted from Fig. 132 reported in [10] (open crosses). Lines show results from HSD [36] and UrQMD2.3 [37] calculations as well as from the core-corona model (the latter for the $\mathrm{Pb}+\mathrm{Pb}$ collision system only [39]). The thin vertical bars indicate the statistical errors, which are mostly not visible. The thick shaded bars indicate the systematic errors.

Figure 11

Kaon yields divided by mean number of pions $\langle {\pi }^{\pm }\rangle =0.5(\langle {\pi }^{-}\rangle +\langle {\pi }^{+}\rangle )$ as a function of centrality. Closed symbols show results from centrality selected $\mathrm{Pb}+\mathrm{Pb}$ collisions; open symbols show $p+p$ data from NA49 [9, 10] as well as $\mathrm{C}+\mathrm{C}$, $\mathrm{Si}+\mathrm{Si}$ (at 158$A$ GeV [8]), and central $\mathrm{Pb}+\mathrm{Pb}$ collisions from [2, 3]. The stars denote the values obtained from parametrization “B” in Ref. [38]. Lines show results from HSD [36] and UrQMD2.3 [37] simulations and the core-corona model (the latter for the $\mathrm{Pb}+\mathrm{Pb}$ collision system only [39]). The thin vertical bars indicate the statistical errors, which are mostly not visible. The thick shaded bars indicate the systematic errors.

Figure 12

Core fraction obtained in the core-corona model for different collision systems as a function of the number of participating nucleons [39]. The dashed line connects the endpoints of the curves for the different collision systems. $f\left(\langle N_{\text{w}}\rangle \right)$ denotes the fraction of those nucleons which undergo more than one collision.

Figure 13

Centrality dependence of the double ratio of the K${}^{+}$ to ${\pi }^{-}$ multiplicities scaled to the ratios from the corresponding most central measurement minimum-bias $\mathrm{Pb}+\mathrm{Pb}$ (Au+Au) collisions at SIS [11] and AGS [12, 13] energies (a), at SPS energies (b), and at RHIC energies [14, 15] (c). The dashed lines in panels (a) and (c) are meant to guide the eye and are also drawn as a reference in panel (b). Only statistical errors are shown.

Figure 14

Total relative strangeness approximated by $E_{\text{s}}$ for centrality selected minimum-bias $\mathrm{Pb}+\mathrm{Pb}$ collisions at 40$A$ GeV (squares) and 158$A$ GeV (circles) as a function of $\langle N_{\text{w}}\rangle$. The open symbols are from $p+p$ [9, 10], $\mathrm{C}+\mathrm{C}$, $\mathrm{Si}+\mathrm{Si}$ [2], S+S [40], and central $\mathrm{Pb}+\mathrm{Pb}$ [2] at 158$A$ GeV. The filled symbols represent the data of centrality selected $\mathrm{Pb}+\mathrm{Pb}$ collisions at 40$A$ GeV and 158$A$ GeV (this analysis and [6]). The lines indicate calculations within the statistical model under the assumption of proportionality between ensemble volume and $\langle N_{\text{w}}\rangle$ (dashed), with volumes as derived from percolation calculations for small systems (dotted line) and $\mathrm{Pb}+\mathrm{Pb}$ collisions (solid line) [21], as well as calculations based on the core-corona ansatz at 158$A$ GeV (dash-dotted line). Only statistical errors are shown.