Using the Tsallis distribution for hadron spectra in $pp$ collisions: Pions and quarkonia at $\sqrt{s}=5\mathbf{-}13000\mathrm{GeV}$

A thermal model, based on the Tsallis distribution and blast-wave model, is proposed to compute hadron double-differential spectra $d^{2}N/d{p}_{\mathrm{T}}dy$ in $pp$ (also high-energy $p\overline{p}$) collisions. It successfully describes the available experimental data on pion and quarkonia [$\varphi$, $\mathrm{J}/\psi$, $\psi (2\mathrm{S})$, $\mathrm{\Upsilon }$ family] production at energies from $\sqrt{s}=5\mathrm{GeV}$ to the LHC ones. Simple parametrizations for the $\sqrt{s}$ dependence of the model parameters are provided, allowing predictions for the yields of these particles at new collision energies. The model can be used also for the pion Bose-Einstein correlation studies.

Figure 1

Ratios $V_f/V_0$ and $⟨v_r⟩/(0.4v_s)$ depending on $v_s$.

Figure 2

Kinetic freeze-out temperature $T$ depending on $\sqrt{s}$.

Figure 3

Parameter $n$ for $\pi$, $\mathrm{J}/\psi$ and $\mathrm{\Upsilon }(1\mathrm{S})$ depending on $\sqrt{s}$.

Figure 4

Fitting of invariant cross section vs $p_{\mathrm{T}}$ at midrapidity and different $\sqrt{s}$ values for ${\pi }^0$ (left) and ${\pi }^{-}$ (right) production in $pp$ ($p\overline{p}$ at $\sqrt{s}=540\mathrm{GeV}$) collisions. Symbols represent the data points, and the line is the fit function. The ${\pi }^{-}$ data and line at $\sqrt{s}=30.6\mathrm{GeV}$ are multiplied by 0.6 for a better visibility. The ratios data/fit are shown at the bottom.

Figure 5

Fitting of ${\pi }^{-}$ invariant yield vs $p_{\mathrm{T}}$ at midrapidity and different $\sqrt{s}$ values (left) and ${\pi }^{+}$ cross section vs $p_{\mathrm{T}}$ at $\sqrt{s}=30.6\mathrm{GeV}$ and different rapidity values [70] (right). Data and lines at $y=0.6$, 1.0, 1.4 are multiplied by 0.4, 0.2, and 0.1, respectively, for a better separation. Dashed lines show the fit functions at $\sqrt{s}=6.3$ and 900 GeV for $v_s=0$.

Figure 6

Prediction for ${\pi }^{-}$ (full line), ${\pi }^0$ (dotted line), and ${\pi }^{+}$ (dashed line) $p_{\mathrm{T}}$-integrated invariant yields at midrapidity depending on $\sqrt{s}$ and comparison with the available data.

Figure 7

Fitting of $\varphi$ meson cross section vs $p_{\mathrm{T}}$ at midrapidity and different $\sqrt{s}$ values (left) and at $\sqrt{s}=7\mathrm{TeV}$ and different rapidity values [81, 82] (right). The data and line for $\sqrt{s}=2760\mathrm{GeV}$ are multiplied by 2, and the ones for $y=2.53$, 3.25, 3.97 are multiplied by 0.8, 0.6, and 0.4, respectively, for a better separation.

Figure 8

$\varphi$ meson $p_{\mathrm{T}}$-integrated cross section at midrapidity (full line) and forward rapidity of $y=3.25$ for $p_{\mathrm{T}}>0$ (dashed line) and $p_{\mathrm{T}}>1\mathrm{GeV}/c$ (dotted line) vs $\sqrt{s}$. Data are from PHENIX [44], ALICE [79, 80, 82] for midrapidity, and ALICE [80, 83] for forward rapidity at $p_{\mathrm{T}}>1\mathrm{GeV}/c$.

Figure 9

Fitting of inclusive $\mathrm{J}/\psi$ meson cross section vs $p_{\mathrm{T}}$ at different $\sqrt{s}$ values for midrapidity (left) and forward rapidity of $y=3.25$ (right). Data and lines at $\sqrt{s}=13$, 8 TeV are multiplied by 3 for a better visibility.

Figure 10

Fitting of prompt $\mathrm{J}/\psi$ meson cross section vs $p_{\mathrm{T}}$ at $\sqrt{s}=7\mathrm{TeV}$ and different rapidity values. Data and lines at $y=0.6$, 2.0, 2.75, 4.25 are multiplied by 3, 0.6, 0.4, and 0.3, respectively, for a better separation.

Figure 11

Fraction of nonprompt $\mathrm{J}/\psi$ production vs $p_{\mathrm{T}}$ at midrapidity and different $\sqrt{s}$. Data and lines at $\sqrt{s}=13\mathrm{TeV}$, 7 TeV [100, 101, 106], 1.96 TeV [92], and 0.2 TeV [88] are shifted up by 0.3, 0.2, 0.1, and 0, respectively, for a better separation.

Figure 12

Inclusive $\mathrm{J}/\psi$ meson $p_{\mathrm{T}}$-integrated cross section as a function of $y$ at different $\sqrt{s}$ and in comparison with the data.

Figure 13

Inclusive $\mathrm{J}/\psi$ meson $p_{\mathrm{T}}$-integrated cross section as a function of $\sqrt{s}$ at midrapidity (full line) and forward rapidity of $y=3.25$ (dashed line) and comparison with the data [109].

Figure 14

Fitting of the $\psi (2\mathrm{S})$ meson cross section vs $p_{\mathrm{T}}$ at midrapidity and different $\sqrt{s}$ values (left) and at $\sqrt{s}=7\mathrm{TeV}$ and different rapidity values (right). Data and lines are multiplied by the numbers, indicated in parentheses, for a better separation.

Figure 15

Inclusive $\psi (2\mathrm{S})$ meson $p_{\mathrm{T}}$-integrated cross section as a function of $\sqrt{s}$ at midrapidity (full line) and forward rapidity of $y=3.25$ (dashed line) and comparison with the data.

Figure 16

Fitting of $\mathrm{\Upsilon }(1\mathrm{S})$ meson cross section vs $p_{\mathrm{T}}$ at different $\sqrt{s}$ values for midrapidity (left) and forward rapidity of $y=3.25$ (right). Data and lines at $\sqrt{s}=1.8$, 8 TeV are multiplied by 0.4 and 2, respectively, for a better visibility.

Figure 17

$\mathrm{\Upsilon }(1\mathrm{S})$ meson $p_{\mathrm{T}}$-integrated cross section as a function of $y$ at different $\sqrt{s}$ values and comparison with the data.

Figure 18

$\mathrm{\Upsilon }(1\mathrm{S})$ meson $p_{\mathrm{T}}$-integrated cross section as a function of $\sqrt{s}$ at midrapidity (full line) and forward rapidity of $y=3.25$ (dashed line) and comparison with the data.

Figure 19

Fitting of ratios $R^{21}$ and $R^{31}$ vs $p_{\mathrm{T}}$, described in the text for $\mathrm{\Upsilon }(2\mathrm{S})$ and $\mathrm{\Upsilon }(3\mathrm{S})$, at different $\sqrt{s}$ and $y$ values. Forward rapidity data and lines at $\sqrt{s}=8$, 7, 2.76 TeV are shifted up by 0.1, 0.2, and 0.3, respectively, for a better separation.

Figure 20

$\mathrm{{\rm Y}}(1\mathrm{S}+2\mathrm{S}+3\mathrm{S})$ $p_{\mathrm{T}}$-integrated cross section times the dimuon branching fraction $B$ vs $y$ for $pp$ collisions at $\sqrt{s}=200\mathrm{GeV}$ and comparison with the data [114, 115, 116].