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The average transverse momentum
$\left\langle p_{\mathrm{T}} \right\rangle$ of final particles is an important observable in high-energy heavy-ion collision experiments. It reflects the properties of soft hadrons and thermonuclear matter, and it can also be used to deduce the information about the evolution of collision systems. By using the phenomenological linear and power-law functions, we study the dependence of the average transverse momentum$\langle p_{\mathrm{T}}\rangle$ at midrapidity in Au + Au and Pb + Pb collisions from the STAR, PHENIX and ALICE Collaborations on four normalized physical quantities, i.e. the collision centrality, the average number of binary collisions per participant pair$\dfrac{2N_{{\mathrm{coll}}}}{N_{{\mathrm{part}}}}$ , the average pseudorapidity density of charged particles per participant pair$\dfrac{2}{N_{{\mathrm{part}}}}\dfrac{{\mathrm{d}}N_{{\mathrm{ch}}}}{{\mathrm{d}}\eta}$ and the average pseudorapidity density of charged particles per binary collision$\dfrac{1}{N_{{\mathrm{coll}}}}\dfrac{{\mathrm{d}}N_{{\mathrm{ch}}}}{{\mathrm{d}}\eta} $ . The results show that the average transverse momentum$\langle p_{\mathrm{T}} \rangle$ of identified particles exhibits a good linear relationship with collision centrality, and it follows a nice power-law relationship with the average number of binary collisions per participant pair$\dfrac{2N_{{\mathrm{coll}}}}{N_{{\mathrm{part}}}}$ , the average pseudorapidity density of charged particles per participant pair$\dfrac{2}{N_{{\mathrm{part}}}}\dfrac{{\mathrm{d}}N_{{\mathrm{ch}}}}{{\mathrm{d}}\eta}$ , and the average pseudorapidity density of charged particles per binary collision$\dfrac{1}{N_{{\mathrm{coll}}}}\dfrac{{\mathrm{d}}N_{{\mathrm{ch}}}}{{\mathrm{d}}\eta}$ . It is also found that the fitting parameters in the proposed phenomenological functions for the average transverse momentum$\langle p_{\mathrm{T}} \rangle$ with collision centrality and the average number of binary collisions per participant pair follow a power-law function with collision energy, which endows the phenomenological approach with predictive ability. Therefore, the collision centrality and the average number of binary collisions per participant pair are good physical quantities for studying the average transverse momentum of identified particles in high-energy heavy-ion collisions. The results in this study can be used to predict the average transverse momentum of identified particles at other collision energy of which the experimental data are not available so far. The mass ordering of the average transverse momentum of identified particles, i.e.$\text{π}^{-},\;{\mathrm{K}}^{-} $ and$\bar{{\mathrm{p}}}$ , is also discussed and explained by the particle production time related to energy conservation, at a given collision centrality and energy.[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] -
碰撞系统, 碰撞能量 粒子种类 截距$ b_1 /(\mathrm{GeV}\cdot c^{-1}) $ 斜率$ a_1 /(\mathrm{GeV}\cdot c^{-1}) $ $ \chi^2/{\rm NDF} $ $ \text{π}^{-} $ $ 0.382\pm 0.011 $ $ -7.01\times10^{-4} \pm 2.47\times10^{-4} $ 0.311/7 Au+Au, 7.7 GeV $ {\mathrm{K}}^{-} $ $ 0.545 \pm 0.013 $ $ -1.59\times10^{-3}\pm 2.72\times10^{-4} $ 0.337/7 $ \bar{{\mathrm{p}}} $ $ 0.794\pm 0.030 $ $ -4.05\times10^{-3} \pm 6.07\times10^{-4} $ 0.211/7 $ \text{π}^{-} $ $ 0.388\pm 0.011 $ $ -5.39\times10^{-4} \pm 2.50\times10^{-4} $ 0.397/7 Au+Au, 11.5 GeV $ {\mathrm{K}}^{-} $ $ 0.566 \pm 0.016 $ $ -1.46\times10^{-3}\pm 3.47\times10^{-4} $ 0.531/7 $ \bar{{\mathrm{p}}} $ $ 0.815\pm0.036 $ $ -3.87\times10^{-3} \pm 7.24\times10^{-4} $ 0.097/7 $ \text{π}^{-} $ $ 0.397\pm 0.012 $ $ -6.19\times10^{-4} \pm 2.69\times10^{-4} $ 0.238/7 Au+Au, 14.5 GeV $ {\mathrm{K}}^{-} $ $ 0.572 \pm 0.018 $ $ -1.44\times10^{-3}\pm 3.79\times10^{-4} $ 0.323/7 $ \bar{{\mathrm{p}}} $ $ 0.827\pm0.039 $ $ -3.37\times10^{-3} \pm 8.04\times10^{-4} $ 0.122/7 $ \text{π}^{-} $ $ 0.398\pm 0.014 $ $ -5.08\times10^{-4} \pm 3.12\times10^{-4} $ 0.195/7 Au+Au, 19.6 GeV $ {\mathrm{K}}^{-} $ $ 0.578 \pm 0.020 $ $ -1.42\times10^{-3}\pm 4.30\times10^{-4} $ 0.149/7 $ \bar{{\mathrm{p}}} $ $ 0.845\pm0.042 $ $ -3.55\times10^{-3} \pm 8.64\times10^{-4} $ 0.066/7 $ \text{π}^{-} $ $ 0.410\pm 0.014 $ $ -6.08\times10^{-4} \pm 3.19\times10^{-4} $ 0.093/7 Au+Au, 27 GeV $ {\mathrm{K}}^{-} $ $ 0.588 \pm 0.020 $ $ -1.24\times10^{-3}\pm 4.48\times10^{-4} $ 0.179/7 $ \bar{{\mathrm{p}}} $ $ 0.857\pm0.043 $ $ -3.52\times10^{-3} \pm 8.81\times10^{-4} $ 0.134/7 $ \text{π}^{-} $ $ 0.417\pm 0.015 $ $ -5.84\times10^{-4} \pm3.25\times10^{-4} $ 0.151/7 Au+Au, 39 GeV $ {\mathrm{K}}^{-} $ $ 0.615 \pm 0.021 $ $ -1.22\times10^{-3}\pm 4.71\times10^{-4} $ 0.138/7 $ \bar{{\mathrm{p}}} $ $ 0.882\pm 0.054 $ $ -3.46\times10^{-3} \pm 1.11\times10^{-3} $ 0.091/7 $ \text{π}^{-} $ $ 0.409\pm 0.007 $ $ -5.46\times10^{-4} \pm 2.11\times10^{-4} $ 0.755/7 Au+Au, 62.4 GeV $ {\mathrm{K}}^{-} $ $ 0.663\pm0.016 $ $ -1.80\times10^{-3}\pm 3.20\times10^{-4} $ 0.712/7 $ \bar{{\mathrm{p}}} $ $ 0.984\pm 0.025 $ $ -3.87\times10^{-3} \pm 5.46\times10^{-4} $ 0.501/7 $ \text{π}^{-} $ $ 0.400\pm0.009 $ $ -6.57\times10^{-4} \pm 3.24\times10^{-4} $ 0.384/6 Au+Au, 130 GeV $ {\mathrm{K}}^{-} $ $ 0.666 \pm 0.020 $ $ -1.54\times10^{-3} \pm 4.19\times10^{-4} $ 0.478/6 $ \bar{{\mathrm{p}}} $ $ 1.01\pm 0.042 $ $ -3.77\times10^{-3}\pm8.05\times10^{-4} $ 0.275/6 $ \text{π}^{-} $ $ 0.427\pm0.012 $ $ -7.75\times10^{-4} \pm 2.73\times10^{-4} $ 0.234/7 Au+Au, 200 GeV $ {\mathrm{K}}^{-} $ $ 0.720\pm0.033 $ $ -2.18 \times10^{-3} \pm 6.49\times10^{-4} $ 0.145/7 $ \bar{{\mathrm{p}}} $ $ 1.10\pm0.050 $ $ -4.58\times10^{-3}\pm 9.55\times10^{-4} $ 0.222/7 $ \text{π}^{-} $ $ 0.532\pm0.010 $ $ -9.28\times10^{-4} \pm 2.34\times10^{-4} $ 1.099/7 Pb+Pb, 2.76 TeV $ {\mathrm{K}}^{-} $ $ 0.886 \pm 0.017 $ $ -1.95\times10^{-3} \pm 3.80\times10^{-4} $ 0.960/7 $ \bar{{\mathrm{p}}} $ $ 1.40\pm 0.020 $ $ -5.26\times10^{-3}\pm 4.58\times10^{-4} $ 3.124/7 $ \text{π}^{-} $ $ 0.586\pm0.012 $ $ -1.16\times10^{-3} \pm 2.88\times10^{-4} $ 0.707/7 Pb+Pb, 5.02 TeV $ {\mathrm{K}}^{-} $ $ 0.943 \pm 0.008 $ $ -1.84\times10^{-3} \pm 1.93\times10^{-4} $ 6.723/7 $ \bar{{\mathrm{p}}} $ $ 1.50\pm 0.013 $ $ -5.97\times10^{-3}\pm 2.91\times10^{-4} $ 12.752/7 
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碰撞系统, 碰撞能量 粒子种类 系数$ a_2/(\mathrm{GeV}\cdot c^{-1})$ 指数$ b_2 $ $ \chi^2/{\rm NDF} $ $ \text{π}^{-} $ $ 0.330\pm0.019 $ $ 0.118\pm 0.049 $ 0.180/7 Au+Au, 14.5 GeV $ {\mathrm{K}}^{-} $ $ 0.418\pm0.025 $ $ 0.198\pm 0.052 $ 0.235/7 $ \bar{{\mathrm{p}}} $ $ 0.482\pm 0.045 $ $ 0.343\pm 0.082 $ 0.110/7 $ \text{π}^{-} $ $ 0.344\pm0.019 $ $ 0.104\pm 0.040 $ 0.519/7 Au+Au, 62.4 GeV $ {\mathrm{K}}^{-} $ $ 0.462\pm0.021 $ $ 0.214\pm 0.038 $ 0.413/7 $ \bar{{\mathrm{p}}} $ $ 0.566\pm 0.034 $ $ 0.330\pm 0.047 $ 0.352/7 $ \text{π}^{-} $ $ 0.318\pm0.032 $ $ 0.132\pm 0.066 $ 0.375/6 Au+Au, 130 GeV $ {\mathrm{K}}^{-} $ $ 0.481\pm 0.033 $ $ 0.186 \pm0.051 $ 0.448/6 $ \bar{{\mathrm{p}}} $ $ 0.583\pm 0.049 $ $ 0.318\pm 0.067 $ 0.215/6 $ \text{π}^{-} $ $ 0.338\pm0.020 $ $ 0.128 \pm 0.045 $ 0.149/7 Au+Au, 200 GeV $ {\mathrm{K}}^{-} $ $ 0.482\pm0.038 $ $ 0.221\pm 0.065 $ 0.184/7 $ \bar{{\mathrm{p}}} $ $ 0.617\pm 0.050 $ $ 0.322 \pm 0.066 $ 0.304/7 $ \text{π}^{-} $ $ 0.430\pm0.017 $ $ 0.096 \pm 0.024 $ 0.623/7 Pb+Pb, 2.76 TeV $ {\mathrm{K}}^{-} $ $ 0.674 \pm 0.027 $ $ 0.124 \pm 0.024 $ 0.527/7 $ \bar{{\mathrm{p}}} $ $ 0.848\pm 0.029 $ $ 0.227\pm 0.020 $ 1.731/7 $ \text{π}^{-} $ $ 0.460\pm 0.020 $ $ 0.105\pm 0.026 $ 0.405/7 Pb+Pb, 5.02 TeV $ {\mathrm{K}}^{-} $ $ 0.741 \pm 0.014 $ $ 0.105\pm 0.011 $ 3.765/7 $ \bar{{\mathrm{p}}} $ $ 0.889\pm 0.017 $ $ 0.230\pm 0.011 $ 7.564/7 DownLoad:
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碰撞系统, 碰撞能量 粒子种类 系数$ a_3 /(\mathrm{GeV}\cdot c^{-1})$ 指数$ b_3 $ $ \chi^2/{\rm NDF} $ $ \text{π}^{-} $ $ 0.366\pm0.007 $ $ 0.220\pm 0.142 $ 0.263/5 Au+Au, 7.7 GeV $ {\mathrm{K}}^{-} $ $ 0.509\pm 0.008 $ $ 0.418\pm 0.117 $ 0.551/5 $ \bar{{\mathrm{p}}} $ $ 0.700\pm 0.019 $ $ 0.828 \pm 0.200 $ 0.548/5 $ \text{π}^{-} $ $ 0.366\pm0.01 $6 $ 0.195\pm 0.170 $ 0.063/5 Au+Au, 14.5 GeV $ {\mathrm{K}}^{-} $ $ 0.494\pm 0.022 $ $ 0.361\pm 0.174 $ 0.237/5 $ \bar{{\mathrm{p}}} $ $ 0.631\pm 0.044 $ $ 0.689 \pm 0.269 $ 0.235/5 $ \text{π}^{-} $ $ 0.351\pm0.047 $ $ 0.232\pm 0.299 $ 0.105/5 Au+Au, 19.6 GeV $ {\mathrm{K}}^{-} $ $ 0.427\pm 0.057 $ $ 0.590\pm 0.304 $ 0.462/5 $ \bar{{\mathrm{p}}} $ $ 0.473\pm 0.093 $ $ 1.15 \pm 0.465 $ 0.621/5 $ \text{π}^{-} $ $ 0.346\pm0.045 $ $ 0.261\pm 0.254 $ 0.081/5 Au+Au, 27 GeV $ {\mathrm{K}}^{-} $ $ 0.460\pm0.060 $ $ 0.378\pm 0.254 $ 0.123/5 $ \bar{{\mathrm{p}}} $ $ 0.489\pm 0.094 $ $ 0.893 \pm 0.371 $ 0.245/5 $ \text{π}^{-} $ $ 0.333\pm0.070 $ $ 0.290\pm 0.309 $ 0.083/5 Au+Au, 39 GeV $ {\mathrm{K}}^{-} $ $ 0.428\pm 0.090 $ $ 0.472\pm 0.315 $ 0.146/5 $ \bar{{\mathrm{p}}} $ $ 0.405\pm 0.153 $ $ 1.02 \pm 0.546 $ 0.142/5 $ \text{π}^{-} $ $ 0.317\pm0.038 $ $ 0.260\pm 0.136 $ 0.331/6 Au+Au, 62.4 GeV $ {\mathrm{K}}^{-} $ $ 0.357\pm 0.036 $ $ 0.644\pm 0.127 $ 0.662/6 $ \bar{{\mathrm{p}}} $ $ 0.379\pm 0.050 $ $ 0.997 \pm 0.158 $ 0.507/6 $ \text{π}^{-} $ $ 0.290\pm0.042 $ $ 0.257\pm 0.127 $ 0.356/6 Au+Au, 130 GeV $ {\mathrm{K}}^{-} $ $ 0.410\pm 0.045 $ $ 0.388\pm 0.105 $ 0.452/6 $ \bar{{\mathrm{p}}} $ $ 0.440\pm 0.062 $ $ 0.674 \pm 0.142 $ 0.481/6 $ \text{π}^{-} $ $ 0.266\pm0.056 $ $ 0.344 \pm 0.171 $ 0.278/6 Au+Au, 200 GeV $ {\mathrm{K}}^{-} $ $ 0.286\pm0.087 $ $ 0.683\pm 0.247 $ 0.190/6 $ \bar{{\mathrm{p}}} $ $ 0.291\pm 0.089 $ $ 0.989 \pm0.259 $ 0.383/6 $ \text{π}^{-} $ $ 0.325\pm 0.036 $ $ 0.230 \pm 0.058 $ 0.862/7 Pb+Pb, 2.76 TeV $ {\mathrm{K}}^{-} $ $ 0.471\pm0.052 $ $ 0.295\pm 0.058 $ 1.182/7 $ \bar{{\mathrm{p}}} $ $ 0.442\pm0.041 $ $ 0.538\pm 0.048 $ 3.699/7 $ \text{π}^{-} $ $ 0.305\pm0.045 $ $ 0.282\pm 0.071 $ 0.924/7 Pb+Pb, 5.02 TeV $ {\mathrm{K}}^{-} $ $ 0.502\pm 0.030 $ $ 0.272\pm 0.029 $ 8.162/7 $ \bar{{\mathrm{p}}} $ $ 0.373\pm0.023 $ $ 0.602\pm 0.030 $ 20.985/7 DownLoad:
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碰撞系统, 碰撞能量 粒子种类 系数$ a_4 /(\mathrm{GeV}\cdot c^{-1}) $ 指数$ b_4 $ $ \chi^2/{\rm NDF} $ $ \text{π}^{-} $ $ 0.326\pm 0.044 $ $ -0.156\pm 0.128 $ 0.001/5 Au+Au, 14.5 GeV $ {\mathrm{K}}^{-} $ $ 0.400\pm0.056 $ $ -0.290\pm 0.137 $ 0.026/5 $ \bar{{\mathrm{p}}} $ $ 0.425\pm0.094 $ $ -0.547\pm 0.211 $ 0.052/5 $ \text{π}^{-} $ $ 0.357\pm0.021 $ $ -0.185\pm 0.098 $ 0.411/6 Au+Au, 62.4 GeV $ {\mathrm{K}}^{-} $ $ 0.484\pm0.021 $ $ -0.424\pm 0.086 $ 1.108/6 $ \bar{{\mathrm{p}}} $ $ 0.606\pm 0.036 $ $ -0.674 \pm 0.109 $ 1.402/6 $ \text{π}^{-} $ $ 0.331\pm0.026 $ $ -0.391 \pm 0.190 $ 0.347/6 Au+Au, 130 GeV $ {\mathrm{K}}^{-} $ $ 0.519\pm0.025 $ $ -0.503\pm 0.137 $ 0.720/6 $ \bar{{\mathrm{p}}} $ $ 0.665\pm 0.038 $ $ -0.850\pm 0.181 $ 0.384/6 $ \text{π}^{-} $ $ 0.381\pm0.013 $ $ -0.251 \pm 0.124 $ 0.053/6 Au+Au, 200 GeV $ {\mathrm{K}}^{-} $ $ 0.589\pm0.023 $ $ -0.430\pm 0.170 $ 0.420/6 $ \bar{{\mathrm{p}}} $ $ 0.826\pm 0.033 $ $ -0.627\pm 0.174 $ 0.704/6 $ \text{π}^{-} $ $ 0.526\pm 0.009 $ $ -0.171\pm 0.042 $ 0.511/7 Pb+Pb, 2.76 TeV $ {\mathrm{K}}^{-} $ $ 0.874 \pm0.015 $ $ -0.221 \pm 0.043 $ 0.353/7 $ \bar{{\mathrm{p}}} $ $ 1.36 \pm 0.018 $ $ -0.402\pm 0.036 $ 1.187/7 $ \text{π}^{-} $ $ 0.587\pm 0.013 $ $ -0.167\pm 0.041 $ 0.204/7 Pb+Pb, 5.02 TeV $ {\mathrm{K}}^{-} $ $ 0.946\pm 0.008 $ $ -0.169\pm 0.018 $ 1.997/7 $ \bar{{\mathrm{p}}} $ $ 1.52\pm 0.014 $ $ -0.369\pm 0.018 $ 2.886/7 DownLoad:
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59]
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