Scaling properties of fractional momentum loss of high-$p_T$ hadrons in nucleus-nucleus collisions at $\sqrt{s_{NN}}$ from 62.4 GeV to 2.76 TeV

Measurements of the fractional momentum loss $\left(S_{\mathrm{loss}}\equiv \delta p_T/p_T\right)$ of high-transverse-momentum-identified hadrons in heavy-ion collisions are presented. Using ${\pi }^0$ in $\mathrm{Au}+\mathrm{Au}$ and $\mathrm{Cu}+\mathrm{Cu}$ collisions at $\sqrt{s_{{}_{NN}}}=62.4$ and 200 GeV measured by the PHENIX experiment at the Relativistic Heavy Ion Collider and and charged hadrons in $\mathrm{Pb}+\mathrm{Pb}$ collisions measured by the ALICE experiment at the Large Hadron Collider, we studied the scaling properties of $S_{\mathrm{loss}}$ as a function of a number of variables: the number of participants, $N_{\mathrm{part}}$, the number of quark participants, $N_{\mathrm{qp}}$, the charged-particle density, $d{N}_{\mathrm{ch}}/d\eta$, and the Bjorken energy density times the equilibration time, ${\varepsilon }_{\mathrm{Bj}}{\tau }_0$. We find that the $p_T$, where $S_{\mathrm{loss}}$ has its maximum, varies both with centrality and collision energy. Above the maximum, $S_{\mathrm{loss}}$ tends to follow a power-law function with all four scaling variables. The data at $\sqrt{s_{{}_{NN}}}=200$ GeV and 2.76 TeV, for sufficiently high particle densities, have a common scaling of $S_{\mathrm{loss}}$ with $d{N}_{\mathrm{ch}}/d\eta$ and ${\varepsilon }_{\mathrm{Bj}}{\tau }_0$, lending insight into the physics of parton energy loss.

Figure 1

Method of calculating the fractional momentum loss $\left(S_{\mathrm{loss}}\equiv \delta p_T/p_T\right)$. This plot is for illustration only; uncertainties are not shown. The procedure: (1) scale the $p+p$ data by $T_{AA}$ corresponding to the centrality selection of $A+A$ data, (2) fit the $p+p$ data and choose the scaled $p+p$ point closest in yield to the $A+A$ along the fit, and (3) calculate the difference of scaled $p+p$ and $A+A$ transverse momenta, $\delta p_T\equiv p_T^{pp}-p_T^{AA}$, at the same yield.

Figure 2

(a) The number of quark participants as a function of the number of nucleon participants. The error bars represent the systematic uncertainty estimate on the MC-Glauber calculation. The dashed line is a linear fit to the 200-GeV $\mathrm{Au}+\mathrm{Au}$ points with $N_{\mathrm{part}}>100$ to illustrate the nonlinearity of the correlation at low values of $N_{\mathrm{part}}$. (b) The ratio of the number of quark participants to the number of nucleon participants as a function of the number of nucleon participants. The error bands represent the systematic uncertainty estimate on the MC-Glauber calculation. This figure is reproduced from Ref. [20].

Figure 3

$p_T^{pp}$ dependence of $S_{\mathrm{loss}}$ for ${\pi }^0$ in 200-GeV $\mathrm{Au}+\mathrm{Au}$ collisions from (solid symbols) 2007 data [8] and (open symbols) 2004 data from the PHENIX experiment at RHIC for $p_T <10\mathrm{GeV}/c$ [12]. The error boxes corresponding to type- B errors are not shown for year- 2004 data, but the magnitudes are same as the ones for year- 2007 data. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus$ pp norm) are type -C errors and show the absolute amount that the data points would move.

Figure 4

$p_T^{pp}$ dependence of $S_{\mathrm{loss}}$ for ${\pi }^0$ in 200-GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions using the spectra measured by PHENIX at RHIC in 2005 [13]. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) are type C errors and show the absolute amount that the data points would move.

Figure 5

$p_T^{pp}$ dependence of $S_{\mathrm{loss}}$ for ${\pi }^0$ in 62-GeV $\mathrm{Au}+\mathrm{Au}$ collisions using the spectra measured by PHENIX in 2010 [7]. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) are type C errors and show the absolute amount that the data points would move.

Figure 6

$p_T^{pp}$ dependence of $S_{\mathrm{loss}}$ for ${\pi }^0$ in 62.4-GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions using the spectra measured by PHENIX in 2005 [13]. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) are type C errors and show the absolute amount that the data points would move.

Figure 7

$p_T^{pp}$ dependence of $S_{\mathrm{loss}}$ for charged hadrons in 2.76-TeV $\mathrm{Pb}+\mathrm{Pb}$ collisions using the result from the ALICE experiment [16, 19]. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) are type C errors and show the absolute amount that the data points would move.

Figure 8

$p_T^{pp}$ dependence of $S_{\mathrm{loss}}$ for charged pions in 2.76-TeV $\mathrm{Pb}+\mathrm{Pb}$ collisions together with those for charged hadrons from the same collision system. The charged pion result is from the ALICE experiment [17].

Figure 9

$p_T^{pp}$ dependence of $S_{\mathrm{loss}}$ for neutral pions in 2.76-TeV $\mathrm{Pb}+\mathrm{Pb}$ collisions using the result from the ALICE experiment [18].

Figure 10

Scaling variables dependence of $S_{\mathrm{loss}}$ at $p_T^{pp}=7$ GeV/$c$. (a) $S_{\mathrm{loss}}$ vs $N_{\mathrm{part}}$, (b) $S_{\mathrm{loss}}$ vs $N_{\mathrm{qp}}$, (c) $S_{\mathrm{loss}}$ vs $d{N}_{\mathrm{ch}}/d\eta$, and (d) $S_{\mathrm{loss}}$ vs ${\varepsilon }_{\mathrm{Bj}}{\tau }_0$. $N_{\mathrm{qp}}$ are all calculated by PHENIX. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) is not shown in these plots.

Figure 11

Scaling variables dependence of $S_{\mathrm{loss}}$ at $p_T^{pp}=12$ GeV/$c$. (a) $S_{\mathrm{loss}}$ vs $N_{\mathrm{part}}$, (b) $S_{\mathrm{loss}}$ vs $N_{\mathrm{qp}}$, (c) $S_{\mathrm{loss}}$ vs $d{N}_{\mathrm{ch}}/d\eta$, and (d) $S_{\mathrm{loss}}$ vs ${\varepsilon }_{\mathrm{Bj}}{\tau }_0$. $N_{\mathrm{qp}}$ are all calculated by PHENIX. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) is not shown in these plots.

Figure 12

$N_{\mathrm{part}}$ dependence of the fractional momentum loss in bins of $p_T^{pp}$ for various systems and $\sqrt{s_{{}_{NN}}}$. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) is not shown in these plots.

Figure 13

$N_{\mathrm{qp}}$ dependence of the fractional momentum loss in bins of $p_T$ for various systems and $\sqrt{s_{{}_{NN}}}$. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) is not shown in these plots. $N_{\mathrm{qp}}$ are all calculated by PHENIX.

Figure 14

$d{N}_{\mathrm{ch}}/d\eta$ dependence of the fractional momentum loss. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) is not shown in these plots.

Figure 15

${\varepsilon }_{\mathrm{Bj}}{\tau }_0$ dependence of the fractional momentum loss. ${\delta }_{\mathrm{sys}}$($T_{AA} \oplus pp$ norm) is not shown in these plots.

Figure 16

$p_T^{pp}$ dependence of fitting parameters for $S_{\mathrm{loss}}$ vs scaling variables. Open symbols correspond to $\mathrm{Pb}+\mathrm{Pb}$ 2.76 TeV, and closed symbols correspond to $\mathrm{Au}+\mathrm{Au}$ 200 GeV. (a) $\alpha$ vs $p_T^{pp}$ and (b) $\beta$ vs $p_T^{pp}$. The Cu+Cu 200 GeV points are not shown, but instead are tabulated in Table 7.