# Nonextensive Statistics in High Energy Collisions

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## Abstract

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## 1. Introduction

## 2. Momentum Distributions

## 3. Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The double-differential transverse momentum distributions with the fits by (

**a**,

**b**) Equation (1), (

**c**,

**d**) Equation (2), and (

**e**–

**h**) Equation (3), with the parameter ${a}_{o}$ left free (

**e**,

**f**) and ${a}_{o}=1$ (

**g**,

**h**). The data are from proton-proton collisons at the center-of-mass energies, $\sqrt{s}=$ 0.9 and 7 TeV; taken from [27,28].

**Figure 2.**Behavior of the Tsallis exponent, q, temperature, T, and the ratio, $T/(q-1)$, as a function of the particle species. extracted from the fits with different formulas. The relation, $n={(q-1)}^{-1}$, used where necessary (see the text for details). (

**a**) Tsallis exponent, q (

**b**) temperature, T (

**c**) the ratio, $T/(q-1)$.

**Figure 3.**Joint confidence region on the T–q plane in the parameter space, with significance, $\alpha =5\%$. The regions are obtained by using (

**a**) Equation (1), (

**b**) Equation (2), and (

**c**,

**d**) Equation (3), with (

**c**) the parameter ${a}_{o}$ left free to adjust and (

**d**) fixed to the value ${a}_{o}=1$. The results for $\sqrt{s}=$ 7 TeV are indicated by the asterisks and the results for $\sqrt{s}=$ 0.9 TeV are indicated by the solid circles for the centers of the ellipses.

**Figure 5.**

**Left**: the dependence of the chemical potential, $\mu $, on the particle species. The results are for the Cleymans Formula (1) (solid squares) and for the Lorentz invariant cross-section Formula (8) (open circles). The dotted and dashed lines represent the average value for the chemical potential, obtained by the Cleymans formula and the Lorentz invariant cross-section formula, respectively.

**Right**: the covariances in the T–q plane in the parameter space, corresponding to Lorentz invariant cross-section Formula. The results for $\sqrt{s}=$ 7 TeV are indicated by the asterisks and the results for $\sqrt{s}=$ 0.9 TeV are indicated by the solid circles for the centers of the ellipses.

**Table 1.**Best-fit parameters of the Cleymans Formula (1). The data represent proton-proton collisions at the center-of-mass energies, $\sqrt{s}=0.9$ and 7 TeV from Refs. [27,28]. The particle masses are taken from [29]. In the last column, the ${\chi}^{2}$-statistics values are given per the number of degrees of freedom (ndf).

$\sqrt{\mathit{s}}$ (TeV) | Particle | q | T (GeV) | ${\mathit{m}}_{0}$ (GeV/${\mathit{c}}^{2}$) | $\mathit{gV}$ (fm${}^{3}$) | $\mathit{\mu}$ (GeV) | ${\mathit{\chi}}^{2}/\mathbf{ndf}$ |
---|---|---|---|---|---|---|---|

0.9 | ${\pi}^{+}$ | 1.148 ± 0.005 | 0.091 ± 0.001 | 0.139570 | $(4.07\pm 0.04)\times {10}^{3}$ | 0.141 ± 0.002 | 3.67/29 |

0.9 | ${\pi}^{-}$ | 1.145 ± 0.005 | 0.076 ± 0.002 | 0.139570 | $(1.80\pm 0.02)\times {10}^{4}$ | 0.031 ± 0.004 | 2.19/29 |

0.9 | ${K}^{+}$ | 1.176 ± 0.015 | 0.092 ± 0.005 | 0.49368 | $(1.02\pm 0.02)\times {10}^{3}$ | 0.20 ± 0.02 | 5.34/23 |

0.9 | ${K}^{-}$ | 1.16 ± 0.01 | 0.084 ± 0.006 | 0.49368 | $(2.33\pm 0.04)\times {10}^{3}$ | 0.129 ± 0.026 | 3.50/23 |

0.9 | p | 1.16 ± 0.02 | 0.09 ± 0.01 | 0.938272 | 774 ± 16 | 0.44 ± 0.05 | 7.43/21 |

0.9 | $\overline{p}$ | 1.13 ± 0.02 | 0.10 ± 0.01 | 0.938272 | 730 ± 20 | 0.36 ± 0.06 | 7.78/20 |

0.9 | ${\pi}^{0}$ | 1.14 ± 0.03 | 0.08 ± 0.04 | 0.134977 | (1 ± 4) × ${10}^{8}$ | −0.05 ± 0.32 | 0.51/9 |

7 | ${\pi}^{0}$ | 1.148 ± 0.005 | 0.13 ± 0.10 | 0.134977 | (0.5 ± 2.7)$\times {10}^{7}$ | 0.2 ± 0.6 | 0.94/29 |

7 | $\eta $ | 1.15 ± 0.03 | 0.1 ± 0.2 | 0.54751 | (0.2 ± 1.8) × ${10}^{7}$ | 0.1 ± 1.2 | 0.09/9 |

$\sqrt{\mathit{s}}$ (TeV) | Particle | q | T (GeV) | ${\mathit{m}}_{0}$ (GeV/${\mathit{c}}^{2}$) | $\mathit{dN}/\mathit{dy}$ | ${\mathit{\chi}}^{2}$/ndf |
---|---|---|---|---|---|---|

0.9 | ${\pi}^{+}$ | 1.148 ± 0.008 | 0.126 ± 0.003 | 0.139570 | 1.49 ± 0.02 | 3.07/30 |

0.9 | ${\pi}^{-}$ | 1.142 ± 0.008 | 0.128 ± 0.003 | 0.139570 | 1.48 ± 0.02 | 1.84/30 |

0.9 | ${K}^{+}$ | 1.21 ± 0.02 | 0.159 ± 0.009 | 0.49368 | 0.184 ± 0.004 | 5.41/24 |

0.9 | ${K}^{-}$ | 1.19 ± 0.02 | 0.162 ± 0.009 | 0.49368 | 0.182 ± 0.004 | 3.59/24 |

0.9 | p | 1.19 ± 0.03 | 0.17 ± 0.01 | 0.938272 | 0.083 ± 0.002 | 7.43/21 |

0.9 | $\overline{p}$ | 1.14 ± 0.03 | 0.19 ± 0.01 | 0.938272 | 0.079 ± 0.002 | 7.75/21 |

0.9 | ${\pi}^{0}$ | 1.15 ± 0.04 | 0.13 ± 0.05 | 0.134977 | (9 ± 5) × ${10}^{4}$ | 0.47/10 |

7 | ${\pi}^{0}$ | 1.171 ± 0.007 | 0.14 ± 0.01 | 0.134977 | (17 ± 3) × ${10}^{4}$ | 1.17/30 |

7 | $\eta $ | 1.17 ± 0.04 | 0.23 ± 0.05 | 0.54751 | (15 ± 5) × ${10}^{3}$ | 0.09/10 |

$\sqrt{\mathit{s}}$ (TeV) | Particle | q | T (GeV) | ${\mathit{m}}_{0}$ (GeV/${\mathit{c}}^{2}$) | C [(GeV/${\mathit{c})}^{-{\mathit{a}}_{0}-1}$] | ${\mathit{a}}_{0}$ | ${\mathit{\chi}}^{2}$/ndf |
---|---|---|---|---|---|---|---|

0.9 | ${\pi}^{+}$ | 1.15 ± 0.01 | 0.12 ± 0.02 | 0.139570 | 43 ± 18 | 1.1 ± 0.2 | 2.69/29 |

0.9 | ${\pi}^{-}$ | 1.148 ± 0.009 | 0.12 ± 0.01 | 0.139570 | 43 ± 17 | 1.1 ± 0.2 | 1.18 /29 |

0.9 | ${K}^{+}$ | 1.22 ± 0.03 | 0.13 ± 0.04 | 0.49368 | 2 ± 1 | 1.2 ± 0.4 | 5.02/23 |

0.9 | ${K}^{-}$ | 1.20 ± 0.03 | 0.14 ± 0.04 | 0.49368 | 1.94 ± 1.20 | 1.3 ± 0.4 | 3.12/23 |

0.9 | p | 1.13 ± 0.07 | 0.23 ± 0.07 | 0.938272 | 0.24 ± 0.08 | 0.6 ± 0.4 | 6.48/20 |

0.9 | $\overline{p}$ | 1.08 ± 0.07 | 0.27 ± 0.07 | 0.938272 | 0.19 ± 0.06 | 0.6 ± 0.3 | 6.27/20 |

0.9 | ${\pi}^{0}$ | 1.1 ± 0.3 | 0.5 ± 1.1 | 0.134977 | (0.8 ± 2.7) × ${10}^{5}$ | −1 ± 3 | 0.34/9 |

7 | ${\pi}^{0}$ | 1.14 ± 0.02 | 0.09 ± 0.03 | 0.134977 | (3± 4) × ${10}^{7}$ | 2 ± 1 | 0.90/29 |

7 | $\eta $ | 1.17 ± 0.06 | 0.2 ± 0.3 | 0.54751 | (0.7 ± 2.0) × ${10}^{5}$ | 1 ± 3 | 0.09/9 |

G (TeV) | Particle | q | T (GeV) | ${\mathit{m}}_{0}$ (GeV/${\mathit{c}}^{2}$) | C [(GeV/${\mathit{c})}^{-{\mathit{a}}_{0}-1}$] | ${\mathit{\chi}}^{2}$/ndf |
---|---|---|---|---|---|---|

0.9 | ${\pi}^{+}$ | 1.148 ± 0.008 | 0.126 ± 0.003 | 0.139570 | 33.4 ± 0.8 | 3.07/30 |

0.9 | ${\pi}^{-}$ | 1.142 ± 0.008 | 0.128 ± 0.003 | 0.139570 | 32.7 ± 0.7 | 1.84/30 |

0.9 | ${K}^{+}$ | 1.21 ± 0.02 | 0.159 ± 0.009 | 0.49368 | 1.30 ± 0.07 | 5.41/24 |

0.9 | ${K}^{-}$ | 1.19 ± 0.02 | 0.162 ± 0.009 | 0.49368 | 1.30 ± 0.06 | 3.59/24 |

0.9 | p | 1.19 ± 0.03 | 0.17 ± 0.01 | 0.938272 | 0.34 ± 0.02 | 7.43/21 |

0.9 | $\overline{p}$ | 1.14 ± 0.03 | 0.19 ± 0.01 | 0.938272 | 0.31 ± 0.02 | 7.75/21 |

0.9 | ${\pi}^{0}$ | 1.15 ± 0.04 | 0.13 ± 0.05 | 0.134977 | (2 ± 2) × ${10}^{6}$ | 0.47/10 |

7 | ${\pi}^{0}$ | 1.171 ± 0.007 | 0.14 ± 0.01 | 0.134977 | (31 ± 9) × ${10}^{5}$ | 1.17/30 |

7 | $\eta $ | 1.17 ± 0.04 | 0.23 ± 0.05 | 0.54751 | (7 ± 4) × ${10}^{4}$ | 0.09/10 |

$\sqrt{\mathit{s}}$ (TeV) | Particle | q | T (GeV) | ${\mathit{m}}_{0}$ (GeV/${\mathit{c}}^{2}$) | $\mathit{gV}$ (fm${}^{3}$) | $\mathit{\mu}$ (GeV) | ${\mathit{\chi}}^{2}$/ndf |
---|---|---|---|---|---|---|---|

0.9 | ${\pi}^{+}$ | 1.148 ± 0.008 | 0.115 ± 0.003 | 0.139570 | $(3.97\pm 0.06)\times {10}^{3}$ | −0.062 ± 0.009 | 3.07/29 |

0.9 | ${\pi}^{-}$ | 1.142 ± 0.008 | 0.124 ± 0.003 | 0.139570 | $(2.43\pm 0.04)\times {10}^{3}$ | −0.017 ± 0.007 | 1.84/29 |

0.9 | ${K}^{+}$ | 1.21 ± 0.02 | 0.107 ± 0.009 | 0.49368 | 789 ± 20 | 0.08 ± 0.04 | 5.41/23 |

0.9 | ${K}^{-}$ | 1.19 ± 0.02 | 0.111 ± 0.009 | 0.49368 | 825 ± 20 | 0.06 ± 0.04 | 3.60/23 |

0.9 | p | 1.19 ± 0.03 | 0.10 ± 0.01 | 0.938272 | 442 ± 8 | 0.41 ± 0.06 | 7.43/20 |

0.9 | $\overline{p}$ | 1.14 ± 0.03 | 0.11 ± 0.02 | 0.938272 | $(1.03\pm 0.08)\times {10}^{3}$ | 0.23 ± 0.08 | 7.75/20 |

0.9 | ${\pi}^{0}$ | 1.15 ± 0.04 | 0.1 ± 0.2 | 0.134977 | (0.1 ± 1.4) × ${10}^{8}$ | −0.08 ± 1.12 | 0.47/9 |

7 | ${\pi}^{0}$ | 1.171 ± 0.007 | 0.12 ± 0.02 | 0.134977 | (2 ± 3) × ${10}^{7}$ | −0.10 ± 0.15 | 1.17/29 |

7 | $\eta $ | 1.17 ± 0.04 | 0.2 ± 0.4 | 0.54751 | (0.6 ± 8.4) × ${10}^{6}$ | 0.1 ± 2.2 | 0.09/9 |

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**MDPI and ACS Style**

Rocha, L.Q.; Megías, E.; Trevisan, L.A.; Olimov, K.K.; Liu, F.; Deppman, A. Nonextensive Statistics in High Energy Collisions. *Physics* **2022**, *4*, 659-671.
https://doi.org/10.3390/physics4020044

**AMA Style**

Rocha LQ, Megías E, Trevisan LA, Olimov KK, Liu F, Deppman A. Nonextensive Statistics in High Energy Collisions. *Physics*. 2022; 4(2):659-671.
https://doi.org/10.3390/physics4020044

**Chicago/Turabian Style**

Rocha, Lucas Q., Eugenio Megías, Luis A. Trevisan, Khusniddin K. Olimov, Fuhu Liu, and Airton Deppman. 2022. "Nonextensive Statistics in High Energy Collisions" *Physics* 4, no. 2: 659-671.
https://doi.org/10.3390/physics4020044