We study the transverse momentum spectra of
In the last century, scientists predicted that a new state of matter could be produced in relativistic heavy-ion (nucleus-nucleus) collisions or could exist in quark stars owing to high temperature and high density [
The transverse momentum (mass) spectra of particles in final state are an important observation. They play one of the major roles in high energy collisions. Other quantities which also play major roles include, but are not limited to, pseudorapidity (or rapidity) distribution, azimuthal distribution (anisotropic flow), particle ratio, and various correlations [
In this paper, we use two methods, the two-component Erlang distribution and the two-component Schwinger mechanism, to describe the transverse momentum spectra of
We assume that the basic impacting process in high energy collisions is binary parton-parton collision. We have two considerations on the description of violent degree of the collision. A consideration is the mean transverse momentum contributed by each parton. The other one is the string tension between two partons. The former consideration can be studied in the framework of Erlang distribution. The latter one results in the Schwinger mechanism. Considering the wide transverse momentum spectra in experiments, we use the two-component Erlang distribution and the two-component Schwinger mechanism. Generally, the first component describes the region of low transverse momentum, and the second one describes the high transverse momentum region.
The transverse momentum spectra,
Values of parameters and
Figure | Type | Two-component Erlang | Two-component Schwinger | ||||||
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Figure |
2.0 < |
0.540 ± 0.062 | 1.008 ± 0.132 | 1.553 ± 0.086 | 0.519 | 0.813 ± 0.027 | 11.70 ± 0.88 | 54.30 ± 3.67 | 1.814 |
2.5 < |
0.545 ± 0.083 | 1.008 ± 0.107 | 1.533 ± 0.082 | 0.846 | 0.810 ± 0.023 | 11.50 ± 1.20 | 50.00 ± 3.32 | 2.318 | |
3.0 < |
0.552 ± 0.102 | 0.978 ± 0.124 | 1.473 ± 0.078 | 0.775 | 0.817 ± 0.029 | 11.42 ± 1.24 | 47.82 ± 3.98 | 1.882 | |
3.5 < |
0.557 ± 0.096 | 0.887 ± 0.093 | 1.373 ± 0.070 | 0.927 | 0.830 ± 0.020 | 10.85 ± 1.10 | 44.00 ± 3.82 | 0.544 | |
4.0 < |
0.564 ± 0.093 | 0.878 ± 0.120 | 1.283 ± 0.100 | 0.913 | 0.830 ± 0.020 | 10.43 ± 1.12 | 38.00 ± 4.23 | 0.256 | |
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Figure |
2.0 < |
0.563 ± 0.084 | 1.183 ± 0.087 | 1.957 ± 0.094 | 0.226 | 0.713 ± 0.023 | 13.80 ± 3.12 | 65.00 ± 7.40 | 0.880 |
2.5 < |
0.582 ± 0.063 | 1.120 ± 0.093 | 1.886 ± 0.112 | 0.940 | 0.778 ± 0.032 | 15.90 ± 3.78 | 69.50 ± 7.90 | 0.492 | |
3.0 < |
0.605 ± 0.078 | 1.118 ± 0.100 | 1.838 ± 0.098 | 0.922 | 0.786 ± 0.037 | 14.60 ± 3.65 | 66.00 ± 7.50 | 0.474 | |
3.5 < |
0.650 ± 0.068 | 1.300 ± 0.092 | 1.570 ± 0.087 | 0.571 | 0.700 ± 0.028 | 12.80 ± 3.07 | 49.80 ± 6.80 | 0.291 | |
4.0 < |
0.647 ± 0.100 | 1.180 ± 0.140 | 1.498 ± 0.153 | 0.105 | 0.657 ± 0.023 | 11.50 ± 3.05 | 40.30 ± 6.30 | 0.218 | |
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Figure |
2.0 < |
0.610 ± 0.077 | 1.180 ± 0.093 | 1.538 ± 0.102 | 0.738 | 0.785 ± 0.025 | 11.90 ± 2.75 | 50.00 ± 5.90 | 2.545 |
2.5 < |
0.640 ± 0.080 | 1.008 ± 0.100 | 1.580 ± 0.087 | 0.705 | 0.860 ± 0.036 | 12.80 ± 2.93 | 55.00 ± 6.30 | 1.255 | |
3.0 < |
0.650 ± 0.065 | 0.953 ± 0.127 | 1.500 ± 0.076 | 0.711 | 0.853 ± 0.032 | 11.00 ± 2.67 | 49.30 ± 6.10 | 1.353 | |
3.5 < |
0.656 ± 0.072 | 0.902 ± 0.118 | 1.404 ± 0.092 | 1.049 | 0.855 ± 0.027 | 10.80 ± 2.70 | 45.00 ± 5.70 | 0.523 | |
4.0 < |
0.725 ± 0.082 | 1.010 ± 0.120 | 1.255 ± 0.085 | 0.707 | 0.776 ± 0.023 | 10.00 ± 2.63 | 33.20 ± 5.20 | 0.708 | |
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Figure |
2.0 < |
0.715 ± 0.062 | 1.120 ± 0.140 | 1.603 ± 0.117 | 0.842 | 0.793 ± 0.038 | 12.00 ± 3.80 | 50.00 ± 5.60 | 2.816 |
2.5 < |
0.640 ± 0.074 | 1.008 ± 0.093 | 1.580 ± 0.082 | 1.171 | 0.796 ± 0.043 | 12.00 ± 4.20 | 48.00 ± 5.80 | 4.194 | |
3.0 < |
0.596 ± 0.054 | 0.947 ± 0.073 | 1.530 ± 0.078 | 1.108 | 0.847 ± 0.032 | 13.00 ± 3.60 | 52.00 ± 5.20 | 0.789 | |
3.5 < |
0.570 ± 0.067 | 0.930 ± 0.056 | 1.413 ± 0.060 | 1.147 | 0.835 ± 0.034 | 10.80 ± 3.20 | 45.00 ± 5.10 | 0.486 | |
4.0 < |
0.600 ± 0.085 | 0.990 ± 0.065 | 1.253 ± 0.067 | 1.051 | 0.805 ± 0.027 | 10.00 ± 3.10 | 37.00 ± 4.70 | 0.268 | |
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Figure |
2.0 < |
0.870 ± 0.064 | 1.200 ± 0.058 | 1.950 ± 0.117 | 1.196 | 0.823 ± 0.043 | 13.00 ± 3.57 | 57.80 ± 6.40 | 1.478 |
2.5 < |
0.837 ± 0.053 | 1.137 ± 0.053 | 1.852 ± 0.130 | 1.114 | 0.830 ± 0.057 | 12.50 ± 3.05 | 56.00 ± 6.10 | 1.880 | |
3.0 < |
0.883 ± 0.057 | 1.145 ± 0.055 | 1.868 ± 0.132 | 0.982 | 0.858 ± 0.068 | 12.80 ± 3.12 | 54.80 ± 6.10 | 1.930 | |
3.5 < |
0.825 ± 0.045 | 1.068 ± 0.062 | 1.693 ± 0.127 | 1.118 | 0.873 ± 0.057 | 12.80 ± 3.07 | 55.30 ± 6.20 | 2.041 | |
4.0 < |
0.838 ± 0.065 | 1.066 ± 0.057 | 1.478 ± 0.142 | 1.192 | 0.853 ± 0.047 | 11.80 ± 2.98 | 43.00 ± 5.90 | 3.430 | |
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Figure |
2.0 < |
0.788 ± 0.063 | 1.372 ± 0.087 | 2.420 ± 0.190 | 1.545 | 0.757 ± 0.047 | 15.60 ± 3.87 | 75.50 ± 8.20 | 0.763 |
2.5 < |
0.644 ± 0.056 | 1.218 ± 0.092 | 2.080 ± 0.217 | 1.787 | 0.725 ± 0.045 | 14.70 ± 3.73 | 70.00 ± 7.50 | 1.226 | |
3.0 < |
0.795 ± 0.078 | 1.300 ± 0.085 | 2.183 ± 0.204 | 1.279 | 0.782 ± 0.053 | 15.00 ± 3.75 | 70.60 ± 7.70 | 0.831 | |
3.5 < |
0.827 ± 0.053 | 1.295 ± 0.075 | 2.120 ± 0.185 | 1.363 | 0.803 ± 0.049 | 14.60 ± 3.65 | 66.00 ± 7.30 | 1.065 | |
4.0 < |
0.810 ± 0.057 | 1.220 ± 0.083 | 1.853 ± 0.177 | 1.733 | 0.822 ± 0.052 | 13.80 ± 3.58 | 58.30 ± 7.10 | 1.342 | |
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Figure |
2.0 < |
0.802 ± 0.188 | 2.560 ± 0.087 | 3.030 ± 0.190 | 0.945 | 0.510 ± 0.047 | 30.00 ± 4.80 | 132.00 ± 12.00 | 0.230 |
2.5 < |
0.830 ± 0.155 | 2.490 ± 0.073 | 2.740 ± 0.178 | 1.523 | 0.524 ± 0.053 | 32.00 ± 4.80 | 120.00 ± 11.70 | 0.268 | |
3.0 < |
0.705 ± 0.165 | 2.360 ± 0.077 | 2.687 ± 0.173 | 1.653 | 0.553 ± 0.057 | 33.00 ± 5.20 | 115.00 ± 11.30 | 0.303 | |
3.5 < |
0.635 ± 0.157 | 2.276 ± 0.084 | 2.652 ± 0.168 | 1.099 | 0.527 ± 0.050 | 28.00 ± 4.00 | 107.00 ± 10.80 | 0.307 | |
4.0 < |
0.771 ± 0.169 | 1.987 ± 0.078 | 2.292 ± 0.153 | 8.640 | 0.550 ± 0.064 | 26.00 ± 3.80 | 87.00 ± 10.10 | 4.617 | |
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Figure |
2.0 < |
0.513 ± 0.137 | 2.836 ± 0.134 | 2.900 ± 0.157 | 2.844 | 0.530 ± 0.057 | 42.00 ± 7.60 | 140.00 ± 22.00 | 1.192 |
2.5 < |
0.614 ± 0.156 | 2.463 ± 0.129 | 3.537 ± 0.123 | 1.791 | 0.517 ± 0.052 | 36.00 ± 7.30 | 145.00 ± 25.00 | 0.732 | |
3.0 < |
0.570 ± 0.135 | 2.675 ± 0.135 | 3.238 ± 0.137 | 1.715 | 0.503 ± 0.049 | 33.00 ± 7.30 | 150.00 ± 25.00 | 0.726 | |
3.5 < |
0.587 ± 0.143 | 2.587 ± 0.138 | 2.620 ± 0.145 | 3.942 | 0.552 ± 0.062 | 37.00 ± 7.5 | 117.00 ± 20.00 | 2.540 | |
4.0 < |
0.535 ± 0.195 | 2.782 ± 0.128 | 3.583 ± 0.237 | 1.970 | 0.510 ± 0.054 | 48.00 ± 7.80 | 175.00 ± 29.00 | 2.163 | |
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Figure |
2.0 < |
0.555 ± 0.148 | 3.342 ± 0.145 | 3.583 ± 0.207 | 4.141 | 0.447 ± 0.078 | 43.00 ± 9.00 | 185.00 ± 28.00 | 3.373 |
2.5 < |
0.783 ± 0.187 | 3.107 ± 0.137 | 3.506 ± 0.198 | 2.554 | 0.493 ± 0.059 | 45.00 ± 7.00 | 180.00 ± 21.00 | 1.594 | |
3.0 < |
0.603 ± 0.152 | 2.886 ± 0.132 | 3.583 ± 0.212 | 3.949 | 0.402 ± 0.062 | 34.00 ± 8.00 | 140.00 ± 19.00 | 2.585 | |
3.5 < |
0.647 ± 0.160 | 2.285 ± 0.107 | 3.350 ± 0.205 | 13.242 | 0.516 ± 0.057 | 30.00 ± 7.50 | 150.00 ± 20.00 | 8.291 | |
4.0 < |
0.533 ± 0.133 | 3.012 ± 0.153 | 3.627 ± 0.223 | 11.216 | 0.423 ± 0.083 | 40.00 ± 10.70 | 170.00 ± 25.00 | 11.114 | |
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Figure |
1.5 < |
0.797 ± 0.133 | 1.285 ± 0.105 | 1.768 ± 0.242 | 3.624 | 0.687 ± 0.076 | 12.60 ± 3.25 | 52.00 ± 5.70 | 0.472 |
2.0 < |
0.802 ± 0.148 | 1.356 ± 0.094 | 1.752 ± 0.216 | 2.201 | 0.692 ± 0.052 | 13.10 ± 3.30 | 53.10 ± 5.80 | 0.614 | |
2.5 < |
0.873 ± 0.097 | 1.322 ± 0.108 | 1.687 ± 0.198 | 2.842 | 0.708 ± 0.067 | 12.40 ± 3.23 | 52.10 ± 5.80 | 0.243 | |
3.0 < |
0.835 ± 0.085 | 1.280 ± 0.107 | 1.352 ± 0.255 | 3.825 | 0.737 ± 0.054 | 12.00 ± 3.05 | 47.00 ± 5.40 | 0.989 | |
3.5 < |
0.870 ± 0.090 | 1.195 ± 0.095 | 1.342 ± 0.188 | 3.814 | 0.726 ± 0.054 | 10.80 ± 3.10 | 43.00 ± 5.20 | 1.383 | |
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Figure |
1.5 < |
0.812 ± 0.158 | 1.656 ± 0.084 | 1.830 ± 0.190 | 0.918 | 0.518 ± 0.065 | 12.60 ± 3.70 | 57.50 ± 5.50 | 1.921 |
2.0 < |
0.684 ± 0.136 | 1.363 ± 0.082 | 2.497 ± 0.203 | 1.309 | 0.543 ± 0.060 | 12.60 ± 3.57 | 57.60 ± 5.80 | 1.807 | |
2.5 < |
0.850 ± 0.107 | 1.405 ± 0.078 | 2.293 ± 0.238 | 4.805 | 0.634 ± 0.067 | 12.30 ± 3.42 | 56.40 ± 5.30 | 1.401 | |
3.0 < |
0.847 ± 0.123 | 1.285 ± 0.080 | 2.050 ± 0.255 | 9.194 | 0.620 ± 0.070 | 12.20 ± 3.37 | 50.60 ± 5.00 | 2.658 | |
3.5 < |
0.835 ± 0.130 | 1.282 ± 0.065 | 2.261 ± 0.273 | 9.018 | 0.657 ± 0.065 | 11.70 ± 3.15 | 50.30 ± 5.10 | 4.329 | |
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Figure |
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0.702 ± 0.158 | 0.937 ± 0.060 | 0.960 ± 0.195 | 0.778 | 0.567 ± 0.063 | 5.00 ± 0.170 | 17.00 ± 2.30 | 0.012 |
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0.982 ± 0.113 | 1.000 ± 0.102 | 1.200 ± 0.227 | 1.038 | 0.512 ± 0.055 | 5.70 ± 0.183 | 18.60 ± 3.10 | 1.428 | |
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0.823 ± 0.083 | 1.020 ± 0.087 | 1.200 ± 0.253 | 1.163 | 0.518 ± 0.047 | 5.80 ± 0.180 | 19.40 ± 3.42 | 1.274 |
Transverse momentum spectra of (a) prompt
Figures
Transverse momentum spectra of (a) prompt
Transverse momentum spectra of (a)
The same as Figure
The transverse momentum spectra,
Transverse momentum spectra of
To give a comparison of fit quality with some other approach, as an example, we show the result of the Tsallis statistics [
In the above fit to the experimental data of LHCb and ALICE Collaborations, the uncorrelated and correlated uncertainties in experimental data are together included in the calculation of
To see clearly the relationships between parameters and rapidity, parameters and centrality, and parameters and others, we plot the values listed in Table
Values of intercepts, slopes, and
Figure | Type | Intercept | Slope |
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Figure |
Prompt |
0.513 ± 0.001 | 0.012 ± 0.001 | 0.001 |
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0.456 ± 0.021 | 0.047 ± 0.006 | 0.003 | |
Prompt |
0.496 ± 0.035 | 0.049 ± 0.011 | 0.007 | |
Prompt |
0.819 ± 0.065 | −0.060 ± 0.020 | 0.030 | |
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Figure |
Prompt |
0.900 ± 0.045 | −0.015 ± 0.014 | 0.013 |
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0.625 ± 0.135 | 0.045 ± 0.041 | 0.139 | |
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Figure |
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0.916 ± 0.142 | −0.051 ± 0.043 | 0.053 |
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0.553 ± 0.085 | 0.003 ± 0.025 | 0.025 | |
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0.741 ± 0.200 | −0.036 ± 0.060 | 0.111 | |
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Figure |
Prompt |
0.737 ± 0.040 | 0.036 ± 0.014 | 0.008 |
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0.691 ± 0.110 | 0.042 ± 0.039 | 0.051 | |
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Figure |
Inclusive | 0.785 ± 0.143 | 0.001 ± 0.003 | 0.344 |
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Figure |
Prompt |
1.200 ± 0.049 | −0.076 ± 0.015 | 0.007 |
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1.067 ± 0.144 | 0.035 ± 0.043 | 0.054 | |
Prompt |
1.301 ± 0.163 | −0.089 ± 0.049 | 0.070 | |
Prompt |
1.219 ± 0.110 | −0.068 ± 0.033 | 0.047 | |
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Figure |
Prompt |
1.342 ± 0.041 | −0.068 ± 0.012 | 0.008 |
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1.429 ± 0.112 | −0.045 ± 0.034 | 0.034 | |
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Figure |
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3.219 ± 0.128 | −0.272 ± 0.039 | 0.028 |
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2.658 ± 0.315 | 0.003 ± 0.095 | 0.087 | |
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3.890 ± 0.670 | −0.296 ± 0.202 | 0.449 | |
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Figure |
Prompt |
1.428 ± 0.080 | −0.051 ± 0.028 | 0.021 |
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1.853 ± 0.144 | −0.165 ± 0.051 | 0.075 | |
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Figure |
Inclusive | 0.937 ± 0.020 | 0.001 ± 0.001 | 0.008 |
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Figure |
Prompt |
1.898 ± 0.054 | −0.140 ± 0.016 | 0.007 |
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2.552 ± 0.119 | −0.247 ± 0.036 | 0.025 | |
Prompt |
1.938 ± 0.116 | −0.148 ± 0.035 | 0.031 | |
Prompt |
2.039 ± 0.096 | −0.173 ± 0.029 | 0.025 | |
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Figure |
Prompt |
2.485 ± 0.142 | −0.221 ± 0.043 | 0.026 |
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2.842 ± 0.227 | −0.219 ± 0.068 | 0.036 | |
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Figure |
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3.697 ± 0.191 | −0.313 ± 0.057 | 0.025 |
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2.884 ± 0.858 | 0.090 ± 0.258 | 0.511 | |
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3.574 ± 0.229 | −0.014 ± 0.069 | 0.022 | |
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Figure |
Prompt |
2.269 ± 0.151 | −0.250 ± 0.053 | 0.027 |
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1.958 ± 0.442 | 0.083 ± 0.156 | 0.180 | |
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Figure |
Inclusive | 0.985 ± 0.092 | 0.004 ± 0.002 | 0.062 |
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Figure |
Prompt |
0.785 ± 0.009 | 0.011 ± 0.003 | 0.001 |
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0.850 ± 0.096 | −0.038 ± 0.029 | 0.132 | |
Prompt |
0.841 ± 0.087 | −0.005 ± 0.026 | 0.108 | |
Prompt |
0.774 ± 0.047 | 0.013 ± 0.014 | 0.025 | |
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Figure |
Prompt |
0.780 ± 0.027 | 0.021 ± 0.008 | 0.005 |
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0.643 ± 0.041 | 0.042 ± 0.012 | 0.015 | |
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Figure |
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0.479 ± 0.027 | 0.017 ± 0.008 | 0.007 |
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0.526 ± 0.041 | −0.001 ± 0.012 | 0.016 | |
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0.472 ± 0.100 | −0.005 ± 0.030 | 0.106 | |
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Figure |
Prompt |
0.642 ± 0.016 | 0.025 ± 0.006 | 0.003 |
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0.399 ± 0.040 | 0.071 ± 0.014 | 0.017 | |
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Figure |
Inclusive | 0.559 ± 0.023 | −0.001 ± 0.001 | 0.031 |
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Figure |
Prompt |
13.254 ± 0.305 | −0.638 ± 0.092 | 0.002 |
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18.725 ± 2.440 | −1.540 ± 0.733 | 0.037 | |
Prompt |
15.070 ± 1.188 | −1.160 ± 0.357 | 0.012 | |
Prompt |
14.940 ± 1.750 | −1.040 ± 0.526 | 0.021 | |
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Figure |
Prompt |
13.945 ± 0.703 | −0.420 ± 0.211 | 0.004 |
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17.145 ± 0.616 | −0.740 ± 0.185 | 0.002 | |
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Figure |
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37.600 ± 4.512 | −2.400 ± 1.357 | 0.042 |
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30.750 ± 11.613 | 2.600 ± 3.491 | 0.137 | |
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52.050 ± 11.186 | −4.200 ± 3.363 | 0.130 | |
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Figure |
Prompt |
14.765 ± 0.803 | −0.840 ± 0.283 | 0.007 |
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13.490 ± 0.227 | −0.440 ± 0.080 | 0.001 | |
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Figure |
Inclusive | 5.037 ± 0.247 | 0.013 ± 0.006 | 0.111 |
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Figure |
Prompt |
71.714 ± 2.053 | −7.720 ± 0.617 | 0.007 |
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103.035 ± 12.805 | −13.820 ± 3.850 | 0.110 | |
Prompt |
74.840 ± 9.489 | −8.720 ± 2.853 | 0.107 | |
Prompt |
65.250 ± 7.664 | −5.800 ± 2.304 | 0.075 | |
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Figure |
Prompt |
73.075 ± 7.290 | −6.060 ± 2.192 | 0.053 |
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93.040 ± 4.414 | −7.680 ± 1.327 | 0.012 | |
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Figure |
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179.150 ± 8.275 | −20.600 ± 2.488 | 0.019 |
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118.100 ± 41.500 | 8.400 ± 12.477 | 0.185 | |
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204.000 ± 35.512 | −12.000 ± 10.677 | 0.121 | |
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Figure |
Prompt |
62.695 ± 3.633 | −4.820 ± 1.280 | 0.019 |
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66.250 ± 2.734 | −4.280 ± 0.963 | 0.011 | |
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Figure |
Inclusive | 16.785 ± 0.374 | 0.047 ± 0.009 | 0.005 |
Dependence of contribution ratio
The same as Figure
The same as Figure
The same as Figure
The same as Figure
The same as Figure
To discuss the Schwinger mechanism, if the charmonium (
Further, if the produced charmed or bottom quark stays at haphazard at the middle between the two participant partons, the maximum potential energy of the charmed quark staying in the colour field of the two partons is
From Table
From the above discussions, we obtain the following conclusions: The transverse momentum spectra of The related parameters such as the mean transverse momentum In the error range,
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under Grant no. 11575103.